2. Given the differential equation $y'' - y = 1 + e^x$:
a. Solve the homogeneous equation to obtain $y_c$.
b. Give the appropriate guess $y_p$, including any necessary modifications due to $y_c$. DO NOT SOLVE.
3. True or False: If two functions $y_1$ and $y_2$ are linearly independent, then the Wronskian $W(y_1, y_2) \neq 0$.
4. Consider a general differential equation $ay'' + by' + cy = g(x)$, where $a$, $b$, and $c$ are constants.
Indicate for which input functions $g(x)$, undetermined coefficients will NOT work.
(a) $g(x) = e^x \ln x$
(b) $g(x) = \frac{\sin x}{e^x}$
(c) $g(x) = \sin 2x$
(d) $g(x) = x^2 \cos x$
(e) $g(x) = -2x^{-1}e^x$
(f) $g(x) = \frac{e^x}{\sin x}$
