In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved

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

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Textbook Solutions for Elementary Differential Equations

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In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved

Chapter 4 textbook questions

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Textbook Solutions for Elementary Differential Equations

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In Chapter 2 of Churchill 's Operational Mathematics, thefollowing theorem is proved

Chapter 4 textbook questions

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