562 Cf!APTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM- Find .C (t^2 cosh t).
7. Show that C (-e' -1) - =In--. s
t s - 1(1 - cost) s^2
- Show that C t = In 82 + 1 ·
- Find C (tsin bt).
- Find .C ( te"^1 cos bt).
- Find £.-^1 (in ( + \ ).
s - 1) - Find C-^1 (ins~ 1 ).
For Problems 13-18, solve the initial va lue problem.- y" (t ) + 2y' (t) + y (t) = 2e-•, with y(O) = 0 and y' (0) = 1.
- y" (t) + y (t) = 2 sin t , with y (0) = 0 and y' (0) = -1.
- ty" (t) - ty' (t) - y (t) = 0, with y (O) = O.
- ty" (t) + (t _, 1) y' (t) - 2y (t) = 0, with y (0) = O.
- ty" (t) + ty' (t) - y (t) = 0, with y (0) = 0.
- ty" (t) + (t - 1) y' (t) + y (t) = 0 , with y (0) = o.
1 9. Solve the Laguerre equation ty" (t) + (1 - t) y' (t) + y (t) = 0, with y (0) = 1. - Solve the Laguerre equation ty" (t) + (1 - t) y' (t) + 2y (t) = 0, with y (0) = 1.
12.9 Inverting the Laplace Transform
So far, most of the applications utilizing the Laplace transform have involved a
transform (or part of a transform) expressed by
P (s)
y (s) = Q (s)' (1 2-36)where P and Q are polynomials that have no common factors. The inverse
of Y ( s) is found by using its partial fraction representation and referring to
Table 12.2. We now show how the theory of complex variables can be used
systematically to find the partial fraction representation. Theorem 12.19 is an
extension of Example 8. 7 to n linear factors. We leave the proof to you.