Page 257 - Schaum's Outline of Differential Equations

240 CONVOLUTIONS AND THE UNIT STEP FUNCTION [CHAP. 23

Supplementary problems

23.20. Find x * x. 23.21. Find 2 * x.
4x
2
23.22. Find 4x * e *. 23.23. Find e * e^.
23.24. Find x * e*. 23.25. Find x * xe~ . x
23.26. Find 3 * sin 2x. 23.27. Find jc * cos jc.

In Problems 23.28 through 23.35, use convolutions to find the inverse Laplace transforms of the given functions.

23.33. - with F(s) = Us 2 and G(s) = sl(s 2 + 4). Compare with Problem 23.6.
l

23.36. Gra.phf(x) = 2u(x-2)-u(x-4).

23.37. Graph/(*) = u(x - 2) - 2u(x - 3) + u(x - 4).

23.38. Use the unit step function to give an analytic representation for the function graphed in Fig. 23-8.

Fig. 23-8

23.39. Graph/(*) = u(x - it) cos 2(x - it). 23.40. Graph /(*) = •
(
In Problems 23.41 through 23.48, find £{g(x)} for the given functions.