Page 256 - Schaum's Outline of Differential Equations
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CHAP. 23] CONVOLUTIONS AND THE UNIT STEP FUNCTION 239
23.16. Prove that/(;t) * [g(x) + h(x)] =f(x) * g(x) +f(x) * h(x).
23.17. The following equation is called an integral equation of convolution type.
Assuming that the Laplace Transform for y(x) exists, we solve this equation, and the next two examples,
for y(x).
We see that this integral equation can be written as y(x) =x + y(x) * sin x. Taking the Laplace transform ££ of
both sides and applying Theorem 23.2, we have
c
Solving for £(y} yields
x 3
This implies that y(x) = x-\ , which is indeed the solution, as can be verified by direct substitution as follows:
6
23.18. Use Laplace Transforms to solve the integral equation of convolution type:
Here we have y(x) = 2 - y(x) * e*. Continuing as in Problem 23.17, we find that
which gives y(x) = 2 - 2x as the desired solution.
23.19. Use Laplace Transforms to solve the integral equation of convolution type:
4
(
2
3
Noting that y(x) = x + 4 * y(x), we find that £{y} = which gives y(x) = (-l + e * -4x-8x )as
the solution.