Question

Where have I made an error?
I understand that I'm over complicating here, but i was practicing for the sake of practicing rather than solving the problem.

Answer 1

x''+8x'+7x=2e^(-t)
Complimentary:
x''+8x'+7x=0
(D^2+8D+7)x=0
(D+1)(D+7)x=0

(D+7)x=u
(D+1)u=0

du/dt+u=0
u=Ae^(-t)

dx/dt+7x=Ae^(-t)
dx/dt+dh/dt x=Ae^(-t)
Be^ht dx/dt+Be^ht dh/dt x=Be^ht Ae^(-t)
(d(Be^ht x))/dt=Be^ht Ae^(-t)
〖Be〗^ht x=∫▒〖BAe^((h-1)t) dt〗
e^ht x=A(1/(h-1))e^((h-1)t)
x=A(1/(h-1))e^t

7= dh/dt
h=7t

x=A(1/(7t-1))e^t

Answer 2

〖Be〗^ht
=B e^ht