Could someone tell me if I am going in the right direction with this problem?

Given a system represented by the differential equation:
(D^3 + 5D^2 + 12^D + 15)y(t) = (D + 1.5)f(t)

with initial conditions:
y(0)=-2, y'(0)=3, y''(0)=4

and input:
f(t) = sin(2.5*pi*t)

find:
a) the impulse response of the system h(t)
b) the zero-state response
c) the zero-input response
d) the total solution

The steps I plan to take:
a) set f(t) = δ(t) and assume zero initial conditions, giving:
(s^3 + 5s^2 + 12s + 15)Y(s) = (s + 1.5)
=> Y(s) = H(s) = (s + 1.5)/(s^3 + 5s^2 + 12s + 15)
inverse laplace transform gives h(t)

b) H(s)*F(s) = Y(s)
inverse laplace transform gives zero-state solution y(t)

c) (s^3 + 5s^2 + 12s + 15) = 0
use the roots of the characteristic eq. to find the char. modes
use the given initial conditions to solve for the coefficients

d) the total solution is the sum of the zero-state and zero-input (from (b) and (c))


Does this all look okay? I feel like I should know this better, but I botched the last assignment and have had much less confidence in my understanding since. Thanks for looking over this and giving me any feedback.

 

looks correct, when finding the zero input I like using the Laplace table for the derivatives and including the initial conditions when I write out the equation and then from there it is (initial conditions/s^3 + 5s^2 + 12s + 15) which you simply inverse transform.