$rV = \pi x − f + \mu x \frac{\partial V(x)}{\partial x} + 0.5\sigma^2 x^2 \frac{\partial^2 V(x)}{\partial x^2}$
Why is the general solution given by: $V(x) = A_0 + A_1 x + A_2 x^\lambda + A_3 x^\beta$?
I thought that since this is a second order non-homogenous differential equation, that the solution would consist of a particular + general solution, where the general solution would look something like $V(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$ where $r_i$ the roots of the corresponding polynomial.
Could anyone please explain? Thanks in advance.