Apply the translation theorem to find the Laplace transforms of the functions in 1

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

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = t2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = e3t +1

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = cos t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = sinh t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. f(t) = sin2 t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = ..fi + 3t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = 3t5/2 -4t3

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = t -2e3t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = t3/2 - e-lOt

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = 1 + cosh 5t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = sin 2t + cos 2t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = cos2 2t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = sin 3t cos 3t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = (1 + t)3

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = tet

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = t cos 2t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f(t) = sinh2 3t

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 4"s

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = S-3/2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = - S S- 5

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = -s+5

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = -s -4 5

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = S2 +4

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = S2 +

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 4-s2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 25 _ S2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. cannot be copied

Chapter 4: Problem 4 Elementary Differential Equations 6

Derive the transform of f(t) = sin kt by the method used in the text to derive the formula in (16).

Chapter 4: Problem 4 Elementary Differential Equations 6

Derive the transform of f (t) = sinh kt by the method used in the text to derive the formula in (14).

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the tabulated integral f eax e ax cos bx dx = 2--2 (a cos bx + b sin bx) + C a +b to obtain {cos kt} directly from the definition of the Laplace transform.

Chapter 4: Problem 4 Elementary Differential Equations 6

Show that the function f(t) = sin(et2 ) is of exponential order as t -+ +00 but that its derivative is not.

Chapter 4: Problem 4 Elementary Differential Equations 6

Given a > 0, let f(t) = 1 if 0 t < a, f(t) = 0 if t a. First, sketch the graph of the function f, making clear its value at t = a. Then express f in terms of unit step functions to show that (f(t)} = sl (1 - e-as).

Chapter 4: Problem 4 Elementary Differential Equations 6

Given that 0 < a < b, let f(t) = 1 if a t < b, f(t) = 0 if either t < a or t b. First, sketch the graph of the function f, making clear its values at t = a and t = b. Then express f in terms of unit step functions to show that (f(t)} = sl (e-as - e-bs).

Chapter 4: Problem 4 Elementary Differential Equations 6

The unit staircase function is defined as follows: f(t) =n if n-l t < n, n=1 . 2, 3, ... . (a) Sketch the graph of f to see why its name is appropriate. (b) Show that 00 f(t) = L u(t -n) n=O for all t o. (c) Assume that the Laplace transform of the infinite series in part (b) can be taken termwise (it can). Apply the geometric series to obtain the result _ 1 (f(t)} - s(1 _ e-S)

Chapter 4: Problem 4 Elementary Differential Equations 6

(a) The graph of the function f is shown in Fig. 4. 1. 10. Show that f can be written in the form 00 f(t) = L(-l)n u(t -n). n=O (b) Use the method of Problem 39 to show that _ 1 (f(t)} - s(1 + e-S) o 4 5 0 6 --t FIGURE 4.1.10. The graph of the function of Problem 40.

Chapter 4: Problem 4 Elementary Differential Equations 6

The graph of the square-wave function get) is shown in Fig. 4. 1.11. Express g in terms of the function f of Problem 40 and hence deduce that 1 - e-s 1 s {g(t)} = = - tanh - . s(l + e-S) s 2 IL .---.0 .---.0 _ ' 2345' FIGURE 4.1.11. The graph of the function of Problem 41.

Chapter 4: Problem 4 Elementary Differential Equations 6

Given constants a and b, define h(t) for t 0 by h(t) = {a if n - 1 t < n and n is odd; b if n - 1 t < n and n is even. Sketch the graph of h and apply one of the preceding problems to show that a + be-s {h(t)} = . s(l + e-S)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 4x = 0; x(O) = 5, X' (O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 9x = 0; x(O) = 3, X' (O) = 4

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" - x' - 2x = 0; x(O) = 0, X'(O) = 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 8x' + 15x = 0; x(O) = 2, X'(O) = -3

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + x = sin 2t; x(O) = 0 = X'(O)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 4x = cos t ; x(O) = 0 = X'(O)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + x = cos 3t; x(O) = 1, X'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 9x = 1; x (0) = 0 = x' (0)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 4x' + 3x = 1; x(O) = 0 = x'(O)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 3x' + 2x = t; x(O) = 0, x'(O) = 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x' = 2x + y, y' = 6x + 3y; x(O) = 1, yeO) = -2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x' = x + 2y, y' = x + e-' ; x(O) = yeO) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x' + 2y' + x = 0, x' - y' + y = 0; x (0) = 0, yeO) = 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + 2x + 4y = 0, y" + X + 2y = 0; x(O) = yeO) = 0, x'(O) = y'(O) = -1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x" + x' + y' + 2x - y = 0, y" + x' + y' + 4x -2y = 0; x(O) = yeO) = 1, x'(O) = y'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 1 through 16. x' = x+z, y' = x+y, z' = -2x -z; x(O) = 1, yeO) = 0,z(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 24 F(s) = 18. F(s) = ----:-------: s(s- 3)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 25 F(s) = ----:-------: s(s - 3) s(s + 5)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 26 F(s) = S (S 2 + 4)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 27 F(s) = S (S 2 + 9)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 28 F(s) = 2 2 22. F(s) = -"--2-- S (s + 1)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 29 F(s) = -"--2-- S (s + 1) s(s - 9)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 30 F(s) = S 2(S2 - 1)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 31 F(s) = ---- s(s + l)(s + 2)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 1 to derive {sin kt} from the formula for {cos kt }.

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 1 to derive {cosh kt} from the formula for {sinh kt}

Chapter 4: Problem 4 Elementary Differential Equations 6

(a) Apply Theorem 1 to show that n {t ne al } = __ {t n1 e al } . s -a (b) Deduce that {t neal } = n!j(s - a)n+ 1 for n = 1, 2, 3, . . . .

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 1 as in Example 5 to derive the Laplace transforms in Problems 28 through 30. {t cos kt} = (s 2 + k2) 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 1 as in Example 5 to derive the Laplace transforms in Problems 28 through 30. {t sinh kt} = (s2 _ P) 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 1 as in Example 5 to derive the Laplace transforms in Problems 28 through 30. {t cosh kt } = (S2 _ P) 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the results in Example 5 and Problem 28 to show that p _ 1 { 1 } 1 .

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. {u(t -a) } = S- I e-as for a > O

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. f f(t) = I on the interval [a, b] (where 0 < a < b) and f(t) = 0 otherwise, then e-as _ ebs (f(t) } = -

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. If f(t) = ( _ 1)[ 1] is the square-wave function whose graph is shown in Fig. 4.2.9, then 1 s (f(t)} = - tanh -. s 2 (Suggestion: Use the geometric series.)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. If f (t) is the unit on-off function whose graph is shown in Fig. 4.2. 10, then

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. If g et) is the triangular wave function whose graph is shown in Fig. 4.2. 11, then1 s {g(t) } = 2" tanh -. s 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. If f(t) is the sawtooth function whose graph is shown in Fig. 4.2. 12, then 1 e-s (f(t) } = 2" - s s - (1 e -S ) (Suggestion: Note that f'(t) == 1 where it is defined.)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the Laplace transforms of the functions in Problems 1 through 4. f(t) = t4e"t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the Laplace transforms of the functions in Problems 1 through 4. f(t) = t 3/2e-4t

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the Laplace transforms of the functions in Problems 1 through 4. f (t) = e2t sin 3n

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the Laplace transforms of the functions in Problems 1 through 4. f(t) = e-t/2 cos 2 (t - kn

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = -- 6. F(s) = 2s -4

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = 2s -4 (s + 1)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) =s2 + 4s + 4

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = s2+4s +5

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = s2 _ 6s + 25

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = 9s2 _ 12s + 20

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = - 2-s -4

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = - s - s

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = s2 + 7s + 10

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = S3 _ s2 _ 2s

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = s 3 -5s 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = (S2 + s _ 6)2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = - 4- s - 16

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = (s _ 4)4 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = S + 4 5s 2 +4

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = S4 _ 8s2 + 16

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = (s2 + 2s + 2)2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use partial fractions to find the inverse Laplace transforms of the functions in Problems 11 through 22. F(s) = (4s2 _ 4s + 5)2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the factorization to derive the inverse Laplace transforms listed in Problems 23 through 26. -1 { S4 : 3 4a4 } = cosh at cos at

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the factorization to derive the inverse Laplace transforms listed in Problems 23 through 26. - 1 L4: 4a4 } = 2 :2 sinh at sin at

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the factorization to derive the inverse Laplace transforms listed in Problems 23 through 26. - 1 { s4 : 2 4a4 } = 2 (cosh at sin at + sinh at cos at)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use the factorization to derive the inverse Laplace transforms listed in Problems 23 through 26. - 1 { 4 14} = 3 (cosh at sin at - sinh at cos at)

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" + 6x' + 25x = 0; x(O) = 2, x'(O) = 3

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" - 6x' + 8x = 2; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" -4x = 3t; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" + 4x' + 8x = e-t ; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x(3) + x" - 6x' = 0; x(O) = 0, x'(O) = x"(O) = 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. X(4) -x = 0; x(O) = 1, x'(O) = x"(O) = X(3) (0) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. X(4) + x = 0; x(O) = x'(O) = x"(O) = 0, X(3) (0) = 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. X(4) + 13x" + 36x = 0; x(O) = x"(O) = 0, x'(O) = 2, x(3)(0) = -13

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x(4) + 8x" + 16x = 0; x(O) = x'(O) = x"(O) = 0, x(3)(O) = 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. X(4) +2x" +x = e2t ; x(O) = x'(O) = x"(O) = X(3) (0) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" + 4x' + 13x = te-t ; x(O) = 0, x'(O) = 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Use Laplace transforms to solve the initial value problems in Problems 27 through 38. x" + 6x' + 18x = cos 2t; x(O) = 1, x'(O) = -1

Chapter 4: Problem 4 Elementary Differential Equations 6

Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x' (0) = o Suppose that m = 1, k = 9, c = 0, and F(t) = 6 cos 3t. Use the inverse transform given in Eq. (16) to derive the solution x (t) = t sin 3t. Construct a figure that illustrates the resonance that occurs.

Chapter 4: Problem 4 Elementary Differential Equations 6

Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x' (0) = o Suppose that m = 1, k = 9.04, c = 0.4, and F(t) = 6e-t1 5 cos 3t. Derive the solution x(t) = te-tl 5 sin3t. Show that the maximum value of the amplitude function A(t) = te-tl 5 is A(5) = 5/e. Thus (as indicated in Fig. 4.3.5) the oscillations of the mass increase in amplitude during the first 5 s before being damped out as t -+ +00.

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = t, g(t) == 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = t, get) = eat

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = get) = sin t

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = t2, get) = cos t

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = g(t) = eat

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the convolution f(t) * g(t) in Problems 1 through 6. f(t) = eat , get) = eM (a i= b)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = s(s -3

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = S(S2 + 4

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = (S2 + 9)2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = S2(S2 + k2)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = (S2 + 4)2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F (s) = ----:-::----:-------:::: S(S2 + 4s + 5)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = ------=-- (S - 3)(S2 + 1)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = -,-----::- S4 + 5s2 + 4

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) = t sin3t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) =t 2 cos 2t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) = te2I cos 3t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) =te-l sin2 t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) = - 20. (t = ---t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f) 1 - cos 2t 19. f(t) = - 20. (t = --- t t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) = -- 22. f(t) = t

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f(t). f(t) = t t

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) = In -s+2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) = In -2 - s +4

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) = In ---- (s + 2)(s - 3)

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) = tan- I -s+2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) = In (1 +s12)

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse transforms of the functions in Problems 23 through 28. F(s) _s - (S2 + 1)3

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" + (t -2)x' + x = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" + (3t - l)x' + 3x = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" - (4t + l)x' + 2(2t + l)x = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" + 2(t - l)x' - 2x = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" -2x' + tx = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x (0) = O. tx" + (4t -2)x' + (l3t -4)x = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply the convolution theorem to show that - 1 { 1 yS} = 2 {.Ii e-u 2 du = el erf.Jt. (s - l) s yn o (Suggestion: Substitute u = v't .)

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x (0) = x' (0) = o. x" + 4x = f(t); x(t) = - f(t - .) sin 2. d

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x (0) = x' (0) = o. x" + 2x' + x = f(t); x(t) = 11 u-r f(t - .) d

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x (0) = x' (0) = o. x" + 4x' + 13x = f(t); x(t) = - f(t - .)e2 r sin 3. d.

Chapter 4: Problem 4 Elementary Differential Equations 6

In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved. Suppose that f(t) is continuous for t ;?; 0, that f (t) is of exponential order as t +00, and that 00 an F(s) = L s n+k+1 n=O where 0 k < 1 and the series converges absolutely for s > c. Then 00 ant n+k f(t) = r(n + k + 1) ' Apply this result in Problems 39 through 41. 39. In Example 5 it was shown that C C ( 1 ) - 1 /2 {Jo(t)} = .JS2+T = - 1 + 2" S2 + 1 s s Expand with the aid of the binomial series and then compute the inverse transformation term by term to obtain Finally, note that Jo(O) = 1 implies that C = 1.

Chapter 4: Problem 4 Elementary Differential Equations 6

In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved. Suppose that f(t) is continuous for t ;?; 0, that f (t) is of exponential order as t +00, and that 00 an F(s) = L s n+k+1 n=O where 0 k < 1 and the series converges absolutely for s > c. Then 00 ant n+k f(t) = r(n + k + 1) ' Apply this result in Problems 39 through 41. Expand the function F(s) = S-I/2e-l/s in powers of SI to show that - 1 __el/s } =1_ cos 2.Jt.

Chapter 4: Problem 4 Elementary Differential Equations 6

In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved. Suppose that f(t) is continuous for t ;?; 0, that f (t) is of exponential order as t +00, and that 00 an F(s) = L s n+k+1 n=O where 0 k < 1 and the series converges absolutely for s > c. Then 00 ant n+k f(t) = r(n + k + 1) ' Apply this result in Problems 39 through 41. Show that -1 {eI/s} = Jo (2.Jt) .

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = -S

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) =S 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = --s + 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = ---s - 1se-S

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) =S2 + 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = - S + n

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = S2 + 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = sO - e-2s ) S2 + n2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = S + n

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = S2 +4

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = 2 if 0 t < 3; f(t) = 0 if t 3

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = 1 if 1 t 4; f(t) = 0 if t < 1 or if t > 4

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = sin t if 0 t 2n; J(t) = O if t > 2n

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = cos m if 0 t 2; J(t) = 0 if t > 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = sin t if 0 t 3n; J(t) = O if t > 3n

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = sin 2t if n t 2n; f(t) = 0 if t < n or if t > 2n

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = sin m if 2 t 3; J(t) = O if t < 2 0r if t > 3

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = cos m if 3 t 5; J(t) = O ift < 3 or if t > 5

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = 0 if t < 1; f(t) = t if t 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = t if t 1; J(t) = 1 if t > 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = t if t 1 ; J(t) = 2 - t if 1 t 2; f(t) = 0 if t>2

Chapter 4: Problem 4 Elementary Differential Equations 6

Find the Laplace transforms of the functions given in Problems 11 through 22. J(t) = t 3 if 1 t 2; J(t) = 0 if t < I or if t > 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 with p = 1 to verify that el{ l} = lis

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to verify that el{cos kt } = sl(s2 + k2).

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to show that the Laplace transform of the square-wave function of Fig. 4.5.13 is 1 elU(t) } = s(l + e-a S)

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Theorem 2 to show that the Laplace transform of the sawtooth function J(t) of Fig. 4.5.14 is 1 e -as F(s) = - - . as2 sO -e-as)

Chapter 4: Problem 4 Elementary Differential Equations 6

Let g(t) be the staircase function of Fig. 4.S. 1S. Show that g (t) = (t / a) - f (t), where f is the sawtooth function of Fig. 4.S. 14, and hence deduce that e-as {g(t) } = s(l _ e-a S)

Chapter 4: Problem 4 Elementary Differential Equations 6

Suppose that f (t) is a periodic function of period 2a with f (t) = t if 0 ;;:; t < a and f (t) = 0 if a ;;:; t < 2a. Find {f(t) }

Chapter 4: Problem 4 Elementary Differential Equations 6

Suppose that f(t) is the half-wave rectification of sin kt, shown in Fig. 4.S.16. Show that k {f(t) } = (S 2 + P)(l _ e-"sfk)

Chapter 4: Problem 4 Elementary Differential Equations 6

Let g(t) = u(t - n/k)f(t - n/k), where f(t) is the function of Problem 29 and k > O. Note that h(t) = f(t) + g(t) is the full-wave rectification of sin kt shown in Fig. 4.S. 17. Hence deduce from Problem 29 that k ns {h(t)} = S 2 +k2 coth 2k

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass-spring-dashpot system with external forcing function. Solve the initial value problem mx" + cx' + kx = f(t), x(O) = x'(O) = 0 and construct the graph of the position function x(t). m = 1, k = 4, c = 0; f(t) = 1 if 0 ;;:; t < n, f(t) = 0 if t n

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass-spring-dashpot system with external forcing function. Solve the initial value problem mx" + cx' + kx = f(t), x(O) = x'(O) = 0 and construct the graph of the position function x(t). m = 1, k = 4, c = S; f(t) = 1 if 0 ;;:; t < 2, f(t) = 0 if t 2

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass-spring-dashpot system with external forcing function. Solve the initial value problem mx" + cx' + kx = f(t), x(O) = x'(O) = 0 and construct the graph of the position function x(t). m = 1, k = 9, c = 0; f(t) = sin t if 0 ;;:; t ;;:; 2n, f(t) = 0 if t > 2n

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass-spring-dashpot system with external forcing function. Solve the initial value problem mx" + cx' + kx = f(t), x(O) = x'(O) = 0 and construct the graph of the position function x(t). m = 1, k = 1, c = 0; f(t) = t if 0 ;;:; t < 1, f(t) = 0 if t 1

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass-spring-dashpot system with external forcing function. Solve the initial value problem mx" + cx' + kx = f(t), x(O) = x'(O) = 0 and construct the graph of the position function x(t). m = 1, k = 4, c = 4; f(t) = t if 0 ;;:; t ;;:; 2, f(t) = 0 if t > 2

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 40, the values of the elements of an RLC circuit are given. Solve the initial value problem di 1 l' L- + Ri + - i(r) dr = e(t); dt C 0 with the given impressed voltage e(t). L = 0, R = 100, C = 10-3; e(t) = 100 if 0 ;;:; t < 1; e(t) = 0 if t 1

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 40, the values of the elements of an RLC circuit are given. Solve the initial value problem di 1 l' L- + Ri + - i(r) dr = e(t); dt C 0 with the given impressed voltage e(t). L = 1, R = 0, C = 10-4; e(t) = 100 if 0 ;;:; t < 2n; e(t) = 0 if t 2n

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 40, the values of the elements of an RLC circuit are given. Solve the initial value problem di 1 l' L- + Ri + - i(r) dr = e(t); dt C 0 with the given impressed voltage e(t). L = 1, R = 0, C = 10-4; e(t) = 100 sin lOt if 0;;:; t < n; e(t) = 0 if t n

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 40, the values of the elements of an RLC circuit are given. Solve the initial value problem di 1 l' L- + Ri + - i(r) dr = e(t); dt C 0 with the given impressed voltage e(t). L = 1, R = lS0, C = 2x 10-4; e(t) = lOOt if 0 ;;:; t < I; e(t) = 0 if t 1

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 36 through 40, the values of the elements of an RLC circuit are given. Solve the initial value problem di 1 l' L- + Ri + - i(r) dr = e(t); dt C 0 with the given impressed voltage e(t). L = 1, R = 100, C = 4 x 10-4; e(t) = SOt if 0 ;;:; t < 1 ; e(t) = 0 if t 1

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 41 and 42, a mass-spring-dashpot system with external force f(t) is described. Under the assumption that x(O) = x'(O) = 0, use the method of Example 8 to find the transient and steady periodic motions of the mass. Then construct the graph of the position function x(t). If you would like to check your graph using a numerical DE solver, it may be useful to note that the function f(t) = A[2ut -n)(t -2n)(t -3n) . (t -4n)(t -Sn)(t -6n)) - 1] has the value +A if 0 < t < n, the value -A ifn < t < 2n, and so forth, and hence agrees on the interval [0, 6n] with the square-wave function that has amplitude A and period 2n. (See also the definition of a square-wave function in terms of sawtooth and triangular-wave functions in the application materialfor this section.) m = 1, k = 4, c = 0; f(t) is a square-wave function with amplitude 4 and period 2n.

Chapter 4: Problem 4 Elementary Differential Equations 6

In Problems 41 and 42, a mass-spring-dashpot system with external force f(t) is described. Under the assumption that x(O) = x'(O) = 0, use the method of Example 8 to find the transient and steady periodic motions of the mass. Then construct the graph of the position function x(t). If you would like to check your graph using a numerical DE solver, it may be useful to note that the function f(t) = A[2ut -n)(t -2n)(t -3n) . (t -4n)(t -Sn)(t -6n)) - 1] has the value +A if 0 < t < n, the value -A ifn < t < 2n, and so forth, and hence agrees on the interval [0, 6n] with the square-wave function that has amplitude A and period 2n. (See also the definition of a square-wave function in terms of sawtooth and triangular-wave functions in the application materialfor this section.) m = 1 , k = 10, c = 2; f(t) is a square-wave function with amplitude 10 and period 2n.

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 4x = oCt); x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 4x = o(t) + o(t -n); x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 4x' + 4x = 1 + o(t -2); x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" +2x' +x = t +o(t); x(O) = O, x'(O) = 1

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 2x' + 2x = 20(t -n); x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 9x = o(t - 3n) + cos 3t; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x"+4x'+5x = 0(t-n)+0(t-2n); x(0) = 0, x'(O) = 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Solve the initial value problems in Problems 1 through 8, and graph each solution function x(t). x" + 2x' + x = o(t) -o(t - 2); x(O) = x'(O) = 2

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Duhamel's principle to write an integral formula for the solution of each initial value problem in Problems 9 through 12. x" + 4x = f(t) ; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Duhamel's principle to write an integral formula for the solution of each initial value problem in Problems 9 through 12. x" + 6x' + 9x = f(t); x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Duhamel's principle to write an integral formula for the solution of each initial value problem in Problems 9 through 12. x" + 6x' + 8x = f(t); x(O) = x' (O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

Apply Duhamel's principle to write an integral formula for the solution of each initial value problem in Problems 9 through 12. x" + 4x' + 8x = f(t) ; x(O) = x'(O) = 0

Chapter 4: Problem 4 Elementary Differential Equations 6

This problem deals with a mass m, initially at rest at the origin, that receives an impulse p at time t = O. (a) Find the solution x< (t) of the problem mx" = pdo,< (t) ; x(O) = x'(O) = O. (b) Show that lim x< (t) agrees with the solution of the <-+0 problem mx" = poet) ; x(O) = x'(O) = O. (c) Show that mv = p for t > 0 (v = dx/dt).

Chapter 4: Problem 4 Elementary Differential Equations 6

Verify that u'(t -a) = o(t -a) by solving the problem x' = o(t -a) ; x(O) = 0 to obtain x(t) = u(t -a).

Chapter 4: Problem 4 Elementary Differential Equations 6

This problem deals with a mass m on a spring (with constant k) that receives an impulse Po = mvo at time t = O. Show that the initial value problems and mx" + kx = 0; x(O) = 0, x'(O) = vo mx" + kx = Poo(t); x(O) = 0, x'(O) = 0 have the same solution. Thus the effect of Poo(t) is, indeed, to impart to the particle an initial momentum PO .

Chapter 4: Problem 4 Elementary Differential Equations 6

This is a generalization of Problem 15. Show that the problems ax" + bx' + cx = f(t); x(O) = 0, x'(O) = vo and ax" + bx' + cx = f(t) + avoo(t); x(O) = x'(O) = 0 have the same solution for t > O. Thus the effect of the term avoo(t) is to supply the initial condition x'(O) = vo.

Chapter 4: Problem 4 Elementary Differential Equations 6

Consider an initially passive RC circuit (no inductance) with a battery supplying eo volts. (a) If the switch to the battery is closed at time t = a and opened at time t = b > a (and left open thereafter), show that the current in the circuit satisfies the initial value problem 1 Ri' + C i = eoo(t - a) -eoo(t -b); i(O) = O. (b) Solve this problem if R = 100 n, C = 10- 4 F, eo = 100 V, a = 1 (s), and b = 2 (s). Show that i(t) > 0 if 1 < t < 2 and that i (t) < 0 if t > 2.

Chapter 4: Problem 4 Elementary Differential Equations 6

Consider an initially passive LC circuit (no resistance) with a battery supplying eo volts. (a) If the switch is closed at time t = 0 and opened at time t = a > 0, show that the current in the circuit satisfies the initial value problem 1 Li" + C i = eoo(t) -eoo(t -a); i(O) = i'(O) = O. (b) If L = 1 H, C = 10-2 F, eo = 10 V, and a = n (s), show that ' ( { Sin lOt I t) = o if t < n, if t > n. Thus the current oscillates through five cycles and then stops abruptly when the switch is opened (Fig. 4.6.6).

Chapter 4: Problem 4 Elementary Differential Equations 6

Consider the LC circuit of Problem 1 8(b), except suppose that the switch is alternately closed and opened at times t = 0, rr/lO, 2rr/1O, .... (a) Show that i(t) satisfies the initial value problem

Chapter 4: Problem 4 Elementary Differential Equations 6

Repeat Problem 19, except suppose that the switch is alternately closed and opened at times t = 0, rr/5, 2rr/5, ... , nrr/5, .... Now show that if then nrr (n + l)rr - < t < -'-- -'-- 5 5' i(t) = { in lOt if n is even; if n is odd. Thus the current in alternate cycles of length rr/5 first executes a sine oscillation during one cycle, then is dormant during the next cycle, and so on (see Fig. 4.6.8).

Chapter 4: Problem 4 Elementary Differential Equations 6

Consider an RLC circuit in series with a battery, with L = 1 H, R = 60 Q, C = 10-3 F, and eo = 10 V. (a) Suppose that the switch is alternately closed and opened at times t = 0, rr/lO, 2rr/1O, . ... Show that i (t) satisfies the initial value problem

Chapter 4: Problem 4 Elementary Differential Equations 6

Consider a mass m = 1 on a spring with constant k = 1, initially at rest, but struck with a hammer at each of the instants t = 0, 2rr, 4rr, .... Suppose that each hammer blow imparts an impulse of + 1. Show that the position function x (t) of the mass satisfies the initial value problem

Table of Contents

Textbook Solutions for Elementary Differential Equations

Question

Solution

Apply the translation theorem to find the Laplace transforms of the functions in 1

Chapter 4 textbook questions

Register

Table of Contents

Textbook Solutions for Elementary Differential Equations

Question

Solution

Apply the translation theorem to find the Laplace transforms of the functions in 1

Chapter 4 textbook questions

Login

Register

Reset password