 4.4.6.1: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.1: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.1: Apply the translation theorem to find the Laplace transforms of the...
 4.4.4.1: Find the convolution f(t) * g(t) in 1 through 6. f(t) = t, g(t) == 1
 4.4.1.1: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.1: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.2: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.2: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.2: Apply the translation theorem to find the Laplace transforms of the...
 4.4.4.2: Find the convolution f(t) * g(t) in 1 through 6. f(t) = t, get) = eat
 4.4.1.2: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.2: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.3: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.3: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.3: Apply the translation theorem to find the Laplace transforms of the...
 4.4.4.3: Find the convolution f(t) * g(t) in 1 through 6. f(t) = get) = sin t
 4.4.1.3: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.3: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.4: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.4: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.4: Apply the translation theorem to find the Laplace transforms of the...
 4.4.4.4: Find the convolution f(t) * g(t) in 1 through 6. f(t) = t2, get) = ...
 4.4.1.4: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.4: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.5: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.5: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.5: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.5: Find the convolution f(t) * g(t) in 1 through 6. f(t) = g(t) = eat
 4.4.1.5: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.5: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.6: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.6: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.6: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.6: Find the convolution f(t) * g(t) in 1 through 6. f(t) = eat , get) ...
 4.4.1.6: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.6: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.7: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.7: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.7: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.7: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.1.7: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.5.7: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.6.8: Solve the initial value problems in 1 through 8, and graph each sol...
 4.4.2.8: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.8: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.8: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.8: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.1.8: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.6.9: Apply Duhamel's principle to write an integral formula for the solu...
 4.4.2.9: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.9: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.9: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.9: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.1.9: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.6.10: Apply Duhamel's principle to write an integral formula for the solu...
 4.4.2.10: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.10: Apply the translation theorem to find the inverse Laplace transform...
 4.4.4.10: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.10: Find the inverse Laplace transform f (t) of each function given in ...
 4.4.1.10: Apply the definition in (1) to find directly the Laplace transforms...
 4.4.6.11: Apply Duhamel's principle to write an integral formula for the solu...
 4.4.2.11: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.11: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.11: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.11: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.11: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.12: Apply Duhamel's principle to write an integral formula for the solu...
 4.4.2.12: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.12: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.12: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.12: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.12: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.13: This problem deals with a mass m, initially at rest at the origin, ...
 4.4.2.13: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.13: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.13: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.13: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.13: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.14: Verify that u'(t a) = o(t a) by solving the problem x' = o(t a) ...
 4.4.2.14: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.14: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.14: Apply the convolution theorem to find the inverse Laplace transform...
 4.4.5.14: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.14: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.15: This problem deals with a mass m on a spring (with constant k) that...
 4.4.2.15: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.15: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.15: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.15: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.15: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.16: This is a generalization of 15. Show that the problems ax" + bx' + ...
 4.4.2.16: Use Laplace transforms to solve the initial value problems in 1 thr...
 4.4.3.16: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.16: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.16: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.16: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.1.17: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.17: Consider an initially passive RC circuit (no inductance) with a bat...
 4.4.2.17: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.17: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.17: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.17: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.5.18: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.18: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.18: Consider an initially passive LC circuit (no resistance) with a bat...
 4.4.2.18: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.18: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.18: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.19: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.19: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.19: Consider the LC circuit of 8(b), except suppose that the switch is ...
 4.4.2.19: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.19: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.19: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.20: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.20: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.20: Repeat 19, except suppose that the switch is alternately closed and...
 4.4.2.20: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.20: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.20: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.21: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.21: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.21: Consider an RLC circuit in series with a battery, with L = 1 H, R =...
 4.4.2.21: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.21: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.21: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.22: Find the Laplace transforms of the functions given in 11 through 22...
 4.4.1.22: Use the transforms in Fig. 4.1.2 to find the Laplace transforms of ...
 4.4.6.22: Consider a mass m = 1 on a spring with constant k = 1, initially at...
 4.4.2.22: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.22: Use partial fractions to find the inverse Laplace transforms of the...
 4.4.4.22: In 15 through 22, apply either Theorem 2 or Theorem 3 to find the L...
 4.4.5.23: Apply Theorem 2 with p = 1 to verify that el{ l} = lis
 4.4.1.23: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.23: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.23: Use the factorization to derive the inverse Laplace transforms list...
 4.4.4.23: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.24: Apply Theorem 2 to verify that el{cos kt } = sl(s2 + k2).
 4.4.1.24: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.24: Apply Theorem 2 to find the inverse Laplace transforms of the funct...
 4.4.3.24: Use the factorization to derive the inverse Laplace transforms list...
 4.4.4.24: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.25: Apply Theorem 2 to show that the Laplace transform of the squarewa...
 4.4.1.25: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.25: Apply Theorem 1 to derive {sin kt} from the formula for {cos kt }.
 4.4.3.25: Use the factorization to derive the inverse Laplace transforms list...
 4.4.4.25: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.26: Apply Theorem 2 to show that the Laplace transform of the sawtooth ...
 4.4.1.26: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.26: Apply Theorem 1 to derive {cosh kt} from the formula for {sinh kt}
 4.4.3.26: Use the factorization to derive the inverse Laplace transforms list...
 4.4.4.26: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.27: Let g(t) be the staircase function of Fig. 4.S. 1S. Show that g (t)...
 4.4.1.27: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.27: (a) Apply Theorem 1 to show that n {t ne al } = __ {t n1 e al } . s...
 4.4.3.27: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.27: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.28: Suppose that f (t) is a periodic function of period 2a with f (t) =...
 4.4.1.28: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.28: Apply Theorem 1 as in Example 5 to derive the Laplace transforms in...
 4.4.3.28: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.28: Find the inverse transforms of the functions in 23 through 28. F(s)...
 4.4.5.29: Suppose that f(t) is the halfwave rectification of sin kt, shown i...
 4.4.1.29: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.29: Apply Theorem 1 as in Example 5 to derive the Laplace transforms in...
 4.4.3.29: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.29: In 29 through 34, transform the given differential equation to find...
 4.4.5.30: Let g(t) = u(t  n/k)f(t  n/k), where f(t) is the function of and ...
 4.4.1.30: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.30: Apply Theorem 1 as in Example 5 to derive the Laplace transforms in...
 4.4.3.30: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.30: In 29 through 34, transform the given differential equation to find...
 4.4.5.31: In 31 through 35, the values of mass m, spring constant k, dashpot ...
 4.4.1.31: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.31: Apply the results in Example 5 and to show that p _ 1 { 1 } 1 .
 4.4.3.31: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.31: In 29 through 34, transform the given differential equation to find...
 4.4.5.32: In 31 through 35, the values of mass m, spring constant k, dashpot ...
 4.4.1.32: Use the transforms in Fig. 4.1.2 to find the inverse Laplace transf...
 4.4.2.32: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.32: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.32: In 29 through 34, transform the given differential equation to find...
 4.4.5.33: In 31 through 35, the values of mass m, spring constant k, dashpot ...
 4.4.1.33: Derive the transform of f(t) = sin kt by the method used in the tex...
 4.4.2.33: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.33: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.33: In 29 through 34, transform the given differential equation to find...
 4.4.5.34: In 31 through 35, the values of mass m, spring constant k, dashpot ...
 4.4.1.34: Derive the transform of f (t) = sinh kt by the method used in the t...
 4.4.2.34: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.34: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.34: In 29 through 34, transform the given differential equation to find...
 4.4.5.35: In 31 through 35, the values of mass m, spring constant k, dashpot ...
 4.4.1.35: Use the tabulated integral f eax e ax cos bx dx = 22 (a cos bx + ...
 4.4.2.35: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.35: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.35: Apply the convolution theorem to show that  1 { 1 yS} = 2 {.Ii eu...
 4.4.5.36: In 36 through 40, the values of the elements of an RLC circuit are ...
 4.4.1.36: Show that the function f(t) = sin(et2 ) is of exponential order as ...
 4.4.2.36: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.36: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.36: In 36 through 38, apply the convolution theorem to derive the indic...
 4.4.5.37: In 36 through 40, the values of the elements of an RLC circuit are ...
 4.4.1.37: Given a > 0, let f(t) = 1 if 0 t < a, f(t) = 0 if t a. First, sketc...
 4.4.2.37: Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace ...
 4.4.3.37: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.37: In 36 through 38, apply the convolution theorem to derive the indic...
 4.4.5.38: In 36 through 40, the values of the elements of an RLC circuit are ...
 4.4.1.38: Given that 0 < a < b, let f(t) = 1 if a t < b, f(t) = 0 if either t...
 4.4.3.38: Use Laplace transforms to solve the initial value problems in 27 th...
 4.4.4.38: In 36 through 38, apply the convolution theorem to derive the indic...
 4.4.5.39: In 36 through 40, the values of the elements of an RLC circuit are ...
 4.4.1.39: The unit staircase function is defined as follows: f(t) =n if nl t...
 4.4.3.39: 39 and 40 illustrate two types of resonance in a massspringdashpo...
 4.4.4.39: In Chapter 2 of Churchill 's Operational Mathematics, thefollowing ...
 4.4.5.40: In 36 through 40, the values of the elements of an RLC circuit are ...
 4.4.1.40: (a) The graph of the function f is shown in Fig. 4. 1. 10. Show tha...
 4.4.3.40: 39 and 40 illustrate two types of resonance in a massspringdashpo...
 4.4.4.40: In Chapter 2 of Churchill 's Operational Mathematics, thefollowing ...
 4.4.5.41: In 41 and 42, a massspringdashpot system with external force f(t)...
 4.4.1.41: The graph of the squarewave function get) is shown in Fig. 4. 1.11...
 4.4.4.41: In Chapter 2 of Churchill 's Operational Mathematics, thefollowing ...
 4.4.5.42: In 41 and 42, a massspringdashpot system with external force f(t)...
 4.4.1.42: Given constants a and b, define h(t) for t 0 by h(t) = {a if n  1 ...
Solutions for Chapter 4: Laplace Transform Methods
Full solutions for Elementary Differential Equations  6th Edition
ISBN: 9780132397308
Solutions for Chapter 4: Laplace Transform Methods
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780132397308. Chapter 4: Laplace Transform Methods includes 224 full stepbystep solutions. Since 224 problems in chapter 4: Laplace Transform Methods have been answered, more than 9051 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.