Question

Find the general solution to the equation:
u'' - u = 4 - sin(t) -2e^t

Answer 1

Okay, firstly you want to solve the homogeneous part of the equation (i.e, where u''-u=0).
So do that:
\[\lambda^2-1=0 \rightarrow \lambda = \pm 1\]
So your homogeneous solution is:
\[u_H(t)=c_1e^{-t}+c_2e^{t}\]
Then for the second part use undetermined coefficients.
You want to look for something in the form:
\[y_P(t)=A+B \sin(t)+D \cos(t)+E t e^t\]
NOTE: The t on the e^t comes from the fact that there is an e^t in your homogeneous solution.

Answer 2

From there, you want to differentiation your particular solution twice and then plug it in to solve for you coefficients.
\[y'_P(t)=B \cos(t)-D \sin(t)+E(e^t+te^t)\]
\[y''_P(t)=-B \sin(t)-D \cos(t)+E(e^t+e^t+te^t)\]
Plug this in and solve for the coefficients:
\[-B \sin(t)-D \cos(t)+E(e^t+e^t+te^t)-A-B \sin(t)-D \cos(t)-Ete^t=\]\[4-\sin(t)-2e^t\]
From here group all your terms together using facts from linear algebra so you know that:
-B-B=-1
-D-D=0
E+E=-2
-A=4
So your general solution is the sum of the homogeneous and the particular giving:
\[c_1e^t+c_2e^{-t}-4+\frac{1}{2}\sin(t)-te^t\]
Here is the wolfram page to confirm it:
http://www.wolframalpha.com/input/?i=solve+y''-y%3D4-sin(t)-2e^t