A programmable calculator or a computer will

Elementary Differential Equations | 6th Edition | ISBN: 9780132397308 | Authors: C. Henry Edwards David E. Penney

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = -y, y(O) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. 2. y' = 2y, y(O) = ; y(x) = e2

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = y + 1, y(O) = 1; y(x) = 2ex - 1

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = x - y, y(O) = 1 ; y(x) = 2e-x + x-I

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = y -x-I , y(O) = 1 ; y(x) = 2 + x -eX

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = -2xy, y(O) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = -3x Z y, y(O) = 3; y(x) = 3e-x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. 8. y' = e-Y, y(O) = 0; y(x) = In(x + 1)

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = (l + y Z ), y(O) = 1; y(x) = tan (x + 1l')

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems I through 10, an initial value problem and its exact solution y(x) are given. Apply Euler's method twice to approximate to this solution on the interval [0, H first with step size h = 0.25, then with step size h = 0. 1. Compare the threedecimal-place values of the two approximations at x = with the value y() of the actual solution. y' = 2xy Z , y(O) = 1; y(x) = 1 z

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. 11. y' = y - 2, y(O) = 1; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. y' = (y - 1)z, y(O) = 2; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. yy' = 2x3, y(l) = 3; 1 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. xy' = y Z , y(l) = 1 ; 1 x

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. xy' = 3x -2y, y(2) = 3; 2 x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems I I through 16. In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral mUltiple of 0.2. Throughout, primes denote derivatives with respect to x. y Z y'= 2X5, y(2) = 3; 2 x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 17. y' = x2 + y2, y(O) = 0; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 18. y' = x2 - y2, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 19. y' = x +,JY, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 20. y' = x + , y(O) = -1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 21. y' = In y, y(l) = 2; 1 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 22. y' = x2/3 + y2/3, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 23. y' = sin x + cos y, y(O) = 0; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is requiredfor Problems 17 through 24. In these initial value problems, use Euler's method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y'= 1 : y2 , Y(-1) = 1 ; -1 X 1

Chapter 6: Problem 6 Elementary Differential Equations 6

You bail out of the helicopter of Example 2 and immediately pull the ripcord of your parachute. Now k = 1.6 in Eq. (5), so your downward velocity satisfies the initial value problem dv dt = 32 - 1 .6v, v(O) = 0 (with t in seconds and v in ftlsec). Use Euler's method with a programmable calculator or computer to approximate the solution for 0 t 2, first with step size h = 0.01 and then with h = 0.005, rounding off approximate v-values to one decimal place. What percentage of the limiting velocity 20 ftlsec has been attained after 1 second? After 2 seconds?

Chapter 6: Problem 6 Elementary Differential Equations 6

Suppose the deer population P (t) in a small forest initially numbers 25 and satisfies the logistic equation dP = 0.0225P _ 0.0003P2 dt (with t in months). Use Euler's method with a programmable calculator or computer to approximate the solution for 10 years, first with step size h = 1 and then with h = 0.5, rounding off approximate P-values to integral numbers of deer. What percentage of the limiting population of 75 deer has been attained after 5 years? After 10 years?

Chapter 6: Problem 6 Elementary Differential Equations 6

Use Euler's method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size 6.1 Application h = 0. 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to two decimal places. 27. y' = x2 + y2 - 1, y(O) = 0; y(2) = ?

Chapter 6: Problem 6 Elementary Differential Equations 6

Use Euler's method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size 6.1 Application h = 0. 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to two decimal places. 28. y' = x + y2, y(-2) = 0; y(2) = ?

Chapter 6: Problem 6 Elementary Differential Equations 6

Consider the initial value problem dy 7x- +y=0, y(-l) = 1 . dx (a) Solve this problem for the exact solution 1 y(x) = - X I / 7 ' which has an infinite discontinuity at x = O. (b) Apply Euler's method with step size h = 0. 15 to approximate this solution on the interval -1 x 0.5. Note that, from these data alone, you might not suspect any difficulty near x = O. The reason is that the numerical approximation "jumps across the discontinuity" to another solution of 7xy' + y = 0 for x > O. (c) Finally, apply Euler's method with step sizes h = 0.03 and h = 0.006, but still printing results only at the original points x = -1.00, -0.85, -0.70, ... , 1.20, 1.35. and 1.50. Would you now suspect a discontinuity in the exact solution?

Chapter 6: Problem 6 Elementary Differential Equations 6

Apply Euler's method with successively smaller step sizes on the interval [0, 2] to verify empirically that the solution of the initial value problem dy dx = x2 + i, y(O) = 0 has a vertical asymptote near x = 2.003147. (Contrast this with Example 2, in which y(O) = 1.)

Chapter 6: Problem 6 Elementary Differential Equations 6

The general solution of the equation dy = (l + i) cos x dx is y(x) = tan(C + sin x). With the initial condition y(O) = 0 the solution y(x) = tan(sin x) is well behaved. But with y(O) = 1 the solution y(x) = tan Un + sin x) has a vertical asymptote at x = sinl (n/4) 0.90334. Use Euler's method to verify this fact empirically.

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 1. y' = -y, yeO) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. y' = 2y, yeO) = ! ; y(x) = ! e 2X

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 3. y' = y + 1, yeO) = I; y(x) = 2ex - 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 4. y' = x - y, yeO) = I; y(x) = 2e-x + x-I

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. S. y' = y -x-I , yeO) = 1 ; y(x) = 2 + x -eX

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 6. y' = -2xy, yeO) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. y' = -3x2y, yeO) = 3; y(x) = 3e-x3

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 8. y' = e-Y, yeO) = 0; y(x) = In(x + 1)

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. y' = (l + y2), yeO) = 1 ; y(x) = tan (x + n)

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval [0, 0.5] with step size h = 0. 1. Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points x = 0. 1, 0.2, 0.3, 004, 0.5. 1 10. y' = 2xy2, yeO) = 1 ; y(x) = --l -x2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 11. y' = y -2, y(O) = 1 ; 0 ;' x ;' 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 12. y' = (y - 1)2, y(O) = 2; 0;' x ;' 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 13. yy' = 2x3 , y(l) = 3; 1 ;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 14. xy' = y2, y(l) = 1 ; 1 ;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 15. xy' = 3x -2y, y(2) = 3; 2 ;' x ;' 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.01, then with step size h = 0.005. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. 16. y2y' = 2x5, y(2) = 3; 2;' x ;' 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 17. y' = x2 + y2, y(O) = 0; 0 ;' x ;' 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 18. y' = x2 - y2, y(O) = 1 ; 0;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 19. y' = x + ..;y, y(O) = 1 ; 0 ;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 20. y'=x +-0', y(0) = -1; 0;'x ;, 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 21. y' = In y, y(l) = 2; 1 ;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = X2/3 + y2 ( 3, y(O) = 1 ; 0;' x ;' 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. 23. y' = sin x + cosy, y(O) = 0; 0 ;' x;' 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0. 1, 0.02, 0.004, and 0.0008 to approximate to five decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = -- y(-I) = l ' -1 :s; x :s; 1 1 + y2 '

Chapter 6: Problem 6 Elementary Differential Equations 6

As in Problem 25 of Section 6.1, you bail out of a helicopter and immediately open your parachute, so your downward velocity satisfies the initial value problem dv dt = 32 - 1.6v, v(O) = 0 (with t in seconds and v in ftls). Use the improved Euler method with a programmable calculator or computer to approximate the solution for 0 ;' t ;' 2, first with step size h = 0.01 and then with h = 0.005, rounding off approximate v-values to three decimal places. What percentage of the limiting velocity 20 ftls has been attained after 1 second? After 2 seconds?

Chapter 6: Problem 6 Elementary Differential Equations 6

As in Problem 26 of Section 6. 1, suppose the deer population P(t) in a small forest initially numbers 25 and satisfies the logistic equation dP = 0.0225P _ 0.0003P2 dt (with t in months). Use the improved Euler method with a programmable calculator or computer to approximate the solution for 10 years, first with step size h = 1 and then with h = 0.5, rounding off approximate P-values to three decimal places. What percentage of the limiting population of 75 deer has been attained after 5 years? After 10 years?

Chapter 6: Problem 6 Elementary Differential Equations 6

Use the improved Euler method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size h = 0. 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to four decimal places. y' = x2 + y2 - 1, y(O) = 0; y(2) =

Chapter 6: Problem 6 Elementary Differential Equations 6

Use the improved Euler method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size h = 0. 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to four decimal places. y' = x + y2, y(-2) = 0; y(2) = ?

Chapter 6: Problem 6 Elementary Differential Equations 6

Consider the crossbow bolt of Example 2 in Section 1.8, shot straight upward from the ground with an initial velocity of 49 m/s. Because of linear air resistance, its velocity function v(t) satisfies the initial value problem dv dt = -(0.04)v -9.8, v(O) = 49 with exact solution v(t) = 294e-t/25 - 245. Use a calculator or computer implementation of the improved Euler method to approximate v(t) for 0 ;' t ;' 10 using both n = 50 and n = 100 subintervals. Display the results at intervals of 1 second. Do the two approximationseach rounded to two decimal places-agree both with each other and with the exact solution? If the exact solution were unavailable, explain how you could use the improved Euler method to approximate closely (a) the bolt's time of ascent to its apex (given in Section 1.8 as 4.56 s) and (b) its impact velocity after 9.41 s in the air.

Chapter 6: Problem 6 Elementary Differential Equations 6

Consider now the crossbow bolt of Example 3 in Section 1.8. It still is shot straight upward from the ground with an initial velocity of 49 mis, but because of air resistance proportional to the square of its velocity, its velocity function v(t) satisfies the initial value problem dv - = -(0.001 1)vlvl -9.8, v(O) = 49. dt The symbolic solution discussed in Section 1.8 required separate investigations of the bolt's ascent and its descent, with v(t) given by a tangent function during ascent and by a hyperbolic tangent function during descent. But the improved Euler method requires no such distinction. Use a calculator or computer implementation of the improved Euler method to approximate v(t) for 0 ;' t ;' 10 using both n = 100 and n = 200 subintervals. Display the results at intervals of 1 second. Do the two approximations---each rounded to two decimal placesagree with each other? If an exact solution were unavailable, explain how you could use the improved Euler method to approximate closely (a) the bolt's time of ascent to its apex (given in Section 1.8 as 4.61 s) and (b) its impact velocity after 9.41 s in the air.

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = -y, y(O) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = 2y, y(O) = ; y(x) = e2x

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = y + 1, y(O) = 1; y(x) = 2ex - 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = x - y, y(O) = 1; y(x) = 2e-x + x-I

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = y -x-I , y(O) = 1; y(x) = 2 + x - eX

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = -2xy, y(O) = 2; y(x) = 2e-x

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = -3x2y, y(O) = 3; y(x) = 3e-x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = e-Y, y(O) = 0; y(x) = In(x + 1)

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = (l + y2), y(O) = 1; y(x) = tan (x +]

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval [0, 0.5] with step size h = 0.25. Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points x = 0.25 and 0.5. y' = 2xy2, y(O) = 1 ; y(x) = --l -x2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. y' = y -2, y(O) = 1; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. y' = !(y - 1)2, y(O) = 2; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. yy' = 2x3, y(1) = 3; 1 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. xy' = y2, y(1) = 1 ; 1 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. xy' = 3x -2y, y(2) = 3; 2 x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size h = 0.2, then with step size h = 0. 1. Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for x an integral multiple of 0.2. Throughout, primes denote derivatives with respect to x. y2y' = 2x5, y(2) = 3; 2 x 3

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = x2 + y2, y(O) = 0; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = x2 - y2, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = x +,JY, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = x + ,lfY, y(O) = -1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = In y, y(1) = 2; 1 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = X2/3 + y2/3, y(O) = 1 ; 0 x 2

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = sin x + cosy, y(O) = 0; 0 x 1

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge-Kutta method with step sizes h = 0.2, 0. 1, 0.05, and 0.025 to approximate to six decimal places the values of the solution at five equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to x. y' = -- y(-l) = 1 -1 :::;; x :::;; 1

Chapter 6: Problem 6 Elementary Differential Equations 6

As in Problem 25 of Section 6.2, you bail out of a helicopter and immediately open your parachute, so your downward velocity satisfies the initial value problem dv dt = 32 - 1.6v, v(O) = 0 (with t in seconds and v in ftls). Use the Runge-Kutta method with a programmable calculator or computer to approximate the solution for 0 t 2, first with step size h = 0. 1 and then with h = 0.05, rounding off approximate v-values to three decimal places. What percentage of the limiting velocity 20 ftls has been attained after 1 second? After 2 seconds?

Chapter 6: Problem 6 Elementary Differential Equations 6

As in Problem 26 of Section 6.2, suppose the deer population P(t) in a small forest initially numbers 25 and satisfies the logistic equation dP de = 0.0225P -0.0003P2 (with t in months). Use the Runge-Kutta method with a programmable calculator or computer to approximate the solution for 10 years, first with step size h = 6 and then with h = 3, rounding off approximate P-values to four decimal places. What percentage of the limiting population of 75 deer has been attained after 5 years? After 10 years?

Chapter 6: Problem 6 Elementary Differential Equations 6

Use the Runge-Kutta method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size h = 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to five decimal places. y' = x2 + y2 - 1, y(O) = 0; y(2) =?

Chapter 6: Problem 6 Elementary Differential Equations 6

Use the Runge-Kutta method with a computer system to find the desired solution values in Problems 27 and 28. Start with step size h = 1, and then use successively smaller step sizes until successive approximate solution values at x = 2 agree rounded off to five decimal places. y' = x + !y2, y(-2) = 0; y(2) =?

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems 29 and 30, the linear acceleration a = dv/dt of a moving particle is given by a formula dv/dt = f(t, v), where the velocity v = dy/dt is the derivative of the function y = y(t) giving the position of the particle at time t. Suppose that the velocity v(t) is approximated using the Runge-Kutta method to solve numerically the initial value problem dv dt = f(t, v), v(O) = Vo (19) That is, starting with to = 0 and vo, the formulas in Eqs. (5) and (6) are applied-with t and v in place of x and y-to calculate the successive approximate velocity values VI> Vz, V 3 , ... , Vm at the successive times t" t 2 , t3 , ... ,tm (with tn+' = tn + h). Now suppose that we also want to approximate the distance y(t) traveled by the particle. We can do this by beginning with the initial position y(O) = Yo and calculating (20) (n = 1, 2, 3, ... ), where an = f(tn, vn) v'(tn) is the particle's approximate acceleration at time tn. Theformula in (20) would give the correct increment (from Yn to Yn+') if the acceleration an remained constant during the time interval [tn, tn+,]. Thus, once a table of approximate velocities has been calculated, Eq. (20) provides a simple way to calculate a table of corresponding successive positions. This process is illustrated in the project for this section, by beginning with the velocity data in Fig. 6.3.8 (Example 3) and proceeding tofollow the skydiver's position during her descent to the ground. Consider again the crossbow bolt of Example 2 in Section 1.8, shot straight upward from the ground with an initial velocity of 49 m/s. Because of linear air resistance, its velocity function v = dy/dt satisfies the initial value problem dv dt = -(0.04)v -9.8, v(O) = 49 with exact solution v(t) = 294e-t/25 -245. (a) Use a calculator or computer implementation of the Runge-Kutta method to approximate v(t) for 0 t 10 using both n = 100 and n = 200 subintervals. Display the results at intervals of I second. Do the two approximations--each rounded to four decimal places-agree both with each other and with the exact solution? (b) Now use the velocity data from part (a) to approximate y(t) for 0 :;::; t :;::; 10 using n = 200 subintervals. Display the results at intervals of 1 second. Do these approximate position valueseach rounded to two decimal places-agree with the exact solution y(t) = 7350 (1 - e-t/25) - 245t? (c) If the exact solution were unavailable, explain how you could use the Runge-Kutta method to approximate closely the bolt's times of ascent and descent and the maximum height it attains.

Chapter 6: Problem 6 Elementary Differential Equations 6

In Problems 29 and 30, the linear acceleration a = dv/dt of a moving particle is given by a formula dv/dt = f(t, v), where the velocity v = dy/dt is the derivative of the function y = y(t) giving the position of the particle at time t. Suppose that the velocity v(t) is approximated using the Runge-Kutta method to solve numerically the initial value problem dv dt = f(t, v), v(O) = Vo (19) That is, starting with to = 0 and vo, the formulas in Eqs. (5) and (6) are applied-with t and v in place of x and y-to calculate the successive approximate velocity values VI> Vz, V 3 , ... , Vm at the successive times t" t 2 , t3 , ... ,tm (with tn+' = tn + h). Now suppose that we also want to approximate the distance y(t) traveled by the particle. We can do this by beginning with the initial position y(O) = Yo and calculating (20) (n = 1, 2, 3, ... ), where an = f(tn, vn) v'(tn) is the particle's approximate acceleration at time tn. Theformula in (20) would give the correct increment (from Yn to Yn+') if the acceleration an remained constant during the time interval [tn, tn+,]. Thus, once a table of approximate velocities has been calculated, Eq. (20) provides a simple way to calculate a table of corresponding successive positions. This process is illustrated in the project for this section, by beginning with the velocity data in Fig. 6.3.8 (Example 3) and proceeding tofollow the skydiver's position during her descent to the ground. Now consider again the crossbow bolt of Example 3 in Section 1 .8. It still is shot straight upward from the ground with an initial velocity of 49 mis, but because of air resistance proportional to the square of its velocity, its velocity function v(t) satisfies the initial value problem dt = -(O.OOl l)vlvl - 9.8, v(O) = 49. Beginning with this initial value problem, repeat parts (a) through (c) of Problem 25 (except that you may need n = 200 subintervals to get four-place accuracy in part (a) and n = 400 subintervals for two-place accuracy in part (b. According to the results of Problems 17 and 18 in Section 1 .8, the bolt's velocity and position functions during ascent and descent are given by the following formulas.Ascent: v(t) = (94.388) tan(0.478837 - [0. 103827]t), y(t) = 108.465 + (909.091) In (cos(0.478837 - [0. 103827]t ; Descent: v(t) = -(94.388) tanh(0. 103827[t - 4.61 19]), y(t) = 108.465 - (909.091) In (cosh(O. l 03827[t - 4.61 19]) .

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = x + 2y, x(O) = 0, y' = 2x + y, yeO) = 2; x(t) = e3t - e-t , yet) = e3t + e-t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 2x + 3y, x(O) = 1, y' = 2x + y, yeO) = -1 ; x(t) = e-t , y(t) = -e-t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 3x +4y, x(0) = 1, y' = 3x + 2y, yeO) = 1 ; x(t) = t (8e6t - e-t), yet) = (6e6t + e-t)

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 9x + 5y, x(O) = 1, y' = -6x -2y, yeO) = 0; x(t) = _5e3t + 6e4t , yet) = 6e3t - 6e4t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 2x -5y, x(O) = 2, y' = 4x -2y, yeO) = 3; x(t) = 2cos 4t - sin4t , yet) = 3cos4t + sin 4t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = x - 2y, x(O) = 0, y' = 2x + y, yeO) = 4; x(t) = -4et sin 2t, yet) = 4et cos 2t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 3x - y, x(O) = 2, y' = x + y, yeO) = 1 ; x(t) = (t + 2)e2t , yet) = (t + l)e2t

Chapter 6: Problem 6 Elementary Differential Equations 6

A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h = 0. 1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge-Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = 5x -9y, x (0) = 0, y' = 2x - y, yeO) = -1 ; x(t) = 3e2t sin 3t, yet) = e2t (sin 3t - cos 3t)

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer will be requiredfor the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge-Kutta method with step sizes h = 0. 1 and h = 0.05 to approximate to five decimal places the values x(l) and y(l). Compare the approximations with the actual values. x' = 2x -y, x(O) = 1, y' = x + 2y, yeO) = 0; x(t) = e2t cos t, yet) = e2t sint

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer will be requiredfor the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge-Kutta method with step sizes h = 0. 1 and h = 0.05 to approximate to five decimal places the values x(l) and y(l). Compare the approximations with the actual values. x' = x + 2y, x(O) = 0, y' =x+e-t , y(O) = 0; xCt) = (2e2t -2e-t + 6te-t), yet) = (e2t - e-t + 6te-t)

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer will be requiredfor the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge-Kutta method with step sizes h = 0. 1 and h = 0.05 to approximate to five decimal places the values x(l) and y(l). Compare the approximations with the actual values. x' = -x - y - (1 + t3)e-t , x(O) = 0, y' = -x - y - (t - 3t2)e-t , yeO) = 1 ; x(t) = e-t (sin t - t), y(t) = e-t (cos t + t3)

Chapter 6: Problem 6 Elementary Differential Equations 6

A computer will be requiredfor the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge-Kutta method with step sizes h = 0. 1 and h = 0.05 to approximate to five decimal places the values x(l) and y(l). Compare the approximations with the actual values. x" + x = sin t, x (0) = 0; x(t) = (sin t - t cos t)

Chapter 6: Problem 6 Elementary Differential Equations 6

Suppose that a crossbow bolt is shot straight upward with initial velocity 288 ftl s. If its deceleration due to air resistance is (0.04)v, then its height x(t) satisfies the initial value problem x" = -32 - (0.04)x' ; x(O) = 0, x'(O) = 288. Find the maximum height that the bolt attains and the time required for it to reach this height.

Chapter 6: Problem 6 Elementary Differential Equations 6

Repeat Problem 13, but assume instead that the deceleration of the bolt due to air resistance is (0.0002)v 2

Chapter 6: Problem 6 Elementary Differential Equations 6

Suppose that a projectile is fired straight upward with initial velocity Vo from the surface of the earth. If air resistance is not a factor, then its height x(t) at time t satisfies the initial value problem gR2 . (x + R)2 ' x(O) = 0, x'(O) =vo. Use the values g = 32. 15 ft/s2 0.006089 mi/s2 for the gravitational acceleration of the earth at its surface and R = 3960 mi as the radius of the earth. If Vo = 1 mils, find the maximum height attained by the projectile and its time of ascent to this height.

Chapter 6: Problem 6 Elementary Differential Equations 6

Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160ft Is and air resistance coefficient c = 0.0025. Find the range-the horizontal distance the ball travels before it hits the ground-and its total time of flight with initial inclination angles 40, 45, and 50.

Chapter 6: Problem 6 Elementary Differential Equations 6

Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160ft Is and air resistance coefficient c = 0.0025. Find (to the nearest degree) the initial inclination that maximizes the range. If there were no air resistance it would be exactly 45, but your answer should be less than 4SO

Chapter 6: Problem 6 Elementary Differential Equations 6

Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160ft Is and air resistance coefficient c = 0.0025. Find (to the nearest half degree) the initial inclination angle greater than 45 for which the range is 300 ft.

Chapter 6: Problem 6 Elementary Differential Equations 6

Find the initial velocity of a baseball hit by Babe Ruth (with c = 0.0025 and initial inclination 40) if it hit the bleachers at a point 50 ft high and 500 horizontal feet from home plate.

Chapter 6: Problem 6 Elementary Differential Equations 6

Consider the crossbow bolt of Problem 14, fired with the same initial velocity of 288 ftl s and with the air resistance deceleration (0.0002)v 2 directed opposite its direction of motion. Suppose that this bolt is fired from ground level at an initial angle of 45. Find how high vertically and how far horizontally it goes, and how long it remains in the air.

Chapter 6: Problem 6 Elementary Differential Equations 6

Suppose that an artillery projectile is fired from ground level with initial velocity 3000 ft/s and initial inclination angle 40. Assume that its air resistance deceleration is (0.0001)v2. (a) What is the range of the projectile and what is its total time of flight? What is its speed at impact with the ground? (b) What is the maximum altitude of the projectile, and when is that altitude attained? (c) You will find that the projectile is still losing speed at the apex of its trajectory. What is the minimum speed that it attains during its descent?

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Textbook Solutions for Elementary Differential Equations

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Textbook Solutions for Elementary Differential Equations

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