624 ANSWERS- x(t) =- c t+ 2te-t, and y(t) = te- t.
2 5. x(t) = -2 + 6e-t cos2t + 6e-t sin 2t, and y(t) = - 1 + 3e- t coo2t.
Section 12. 10. Convolution: p age 578
- f(t) = t, g(t) = t and(!· g)(t) = J; f(r)g(t- r)dr = J; r(t - r)dr = ~·
- f(t) =et, g(t) = e^2 t and (g · f)(t) = J; e^2 "et-"dr =-et+ e^21.
- f (t) = c-^1 ( <--dcs-2)) = - 2e^1 + 2e^2 t.
1. f (t) = c-^1 C.c.i+i» = 1 -cost.
f·(g+h) = J; J(r)(g+h)(t-r)dr = J; J(r}g(t-r}dr+ J; f(r)h(t-r)dr =
f ·g+f ·g.
f(t) = c-^1 (.~1) = e^1 + o(t).
y(t) = -tcost+sint.
1 5. C (I; e_,. cos (t -r) dr) = C (et) C (cost) = (a+i)(.2+ t ).
17. Given F (s) = C (! (t)). G(s) = C ( 1) = ~ and g(t) = 1, we have
c-^1 ( F~s)) = F(s)G(s) = (! · g)(t) = J; f (r) g(t - r)dr = J; f (r) dr.
19. F(s) = ,2. 2 and f (t) = e^2 t.
21. F(s) = •• ~ 1 and f(t) = sinht -sint.
23. y(t) = te- t.25. y (t) = (- e^3 -^3 t + e^1 -^1 )U 1 (t).
- y (t) = -2lte-^1 - 119e-t + 120 - 96t + 36t^2 - 8t^3 + t^4 •