Page 119 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

INTEGRAL TRANSFORMS: THE LAPLACE TRANSFORM 109
is

2cos(2t)

and the inverse of
−3 3 2
=−
2
2
s + 4 2 (s + 4)
is
3
− sin(2t)
2
Thus
3
f (t) = 2e −t cos(2t) − e −t sin(2t)
2
Tables of Laplace transforms and inverse transforms can be found in many books such as the
book by Arpaci and in the Schaum’s Outline referenced below. A brief table is given here in
Appendix A.

Problems
1. Solve the problem

y − 2y + 5y = 0

y(0) = y (0) = 0 y (0) = 1
using Laplace transforms.
2. Find the general solution using Laplace transforms
2
y + k y = a

3. Use convolution to find the solution to the following problem for general g(t). Then find
2
the solution for g(t) = t .
y + 2y + y = g(t)

y (0) = y(0) = 0

4. Find the inverse transforms.
s + c
(a) F(s ) =
(s + a)(s + b) 2
1
(b) F(s ) =
(s + a )s 3
2
2