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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 7: Q14E (page 404)

Find the Laplace transform of. $f (t) = \int_{0}^{t} e^{v} s i n (t - v) d v$

Short Answer

Expert verified

The Laplace transformation of $\int_{0}^{t} e^{v} \sin (t - v) d v$ is $\frac{1}{(s^{2} + 1) (s - 1)}$ .

Step by step solution

01

Given that,

$f (t) = \int_{0}^{t} e^{v} \sin (t - v) d v$

02

Applying the Convolution Theorem

$g ⋆ h = \int_{0}^{t} g (t - v) h (v) d v g (t) = \sin t g (s) = \frac{1}{s^{2} + 1} h (t) = e^{t} g (s) = \frac{1}{s - 1}$

03

Step 3: Simplify

$\begin{matrix} L [g ⋆ h] = g (s) \cdot h (s) \\ = \frac{1}{s^{2} + 1} \cdot \frac{1}{s - 1} \\ = \frac{1}{(s^{2} + 1) (s - 1)} \end{matrix}$

Hence,

$f (s) = \frac{1}{(s^{2} + 1) (s - 1)}$

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Most popular questions from this chapter

In Problems $1 - 14$ , solve the given initial value problem using the method of Laplace transforms.

$z'' + 5 z' - 6 z = 21 e^{t - 1}, z (1) = - 1, z' (1) = 9$

In Problems 1-14 , solve the given initial value problem using the method of Laplace transforms.

$w'' + w = t^{2} + 2; w (0) = 1, w' (0) = - 1$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l} x^{'} +y=x; x(0)=0,y(0)=1 \\ {2x}^{'} {+y}^{''} {=u(t-3);y}^{'} (0)=-1 \end{array}$

In Problems $1 - 14$ , solve the given initial value problem using the method of Laplace transforms.

$2 y'' - y' - 2 y = 0$

In problem 15-22, solve the given integral equation or integro differential equation for $y (t)$

$L \int_{0}^{t} (t - v) y (v) d v = 1$

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