12. 10 • CONVOLUTION 577
- EXAMPLE 12.33 Use the convolution method to solve the !VP
y^11 (t) + y (t) = tan t with y (0) = 1 and y' (O) = 2.
Solution We first solve u" (t) +u(t) = 0 with u (0) = 1 and u' (O) = 2. Taking
the Laplace transform yields s^2 U (s) - s - 2 + U (s) = 0. Solving for U (s) gives
U ( s) =
8
2 +
2
1
, and it follows that
s +
u (t) =cost+ 2sin t.
Second, we observe that H ( s) = ~l and h ( t) = si.n t so that
s +
v (t) = (h * g) (t) =lot sin (t -s) tan (s) ds
= [cos (t) In cos.s - sin (t - s)] 1•=t
1 + sms •=O
= cos (t) In
1
cos.t + sin (t).
+smt
