Laplace Transform Additional Problem
Paul Skoufranis
Due Thursday, April 2nd
1) Find the solution to the initial value problem
y 00 − 2y 0 + 2y = 2ex cos(x) + 2ex sin(x),
y(0) = 0,
y 0 (0) = 0,
on the region where x ≥ 0 using the method of undetermined coefficients or variation of parameters (your
choice).
2) Recall Problem 28 in Section 6.2 of the text (which was on sixth assignment) showed that
if
L[f (x)] = F (s)
L[−xf (x)] = F 0 (s).
then
Using this, show that
L[xecx cos(bx)] =
(s − c)2 − b2
((s − c)2 + b2 )2
and
L[xecx sin(bx)] =
2b(s − c)
.
((s − c)2 + b2 )2
Hence
L
−1
(s − c)2 − b2
= xecx cos(bx)
((s − c)2 + b2 )2
and
−1
L
2b(s − c)
= xecx sin(bx).
((s − c)2 + b2 )2
3) Solve the initial value problem in Question (1) by using the Laplace transform along with Question
(2). If you get stuck, the hint on the next page may be useful.
Hint for (3): Note
2s
(2s − 2) + (1 + (s − 1)2 ) − (−1 + (s − 1)2 )
=
.
((s − 1)2 + 1)2
((s − 1)2 + 1)2
2