# Why is the solution to this ODE as follows?

$rV = \pi x − f + \mu x \frac{\partial V(x)}{\partial x} + 0.5\sigma^2 x^2 \frac{\partial^2 V(x)}{\partial x^2}$

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.

This is called a Cauchy–Euler equation. The important point is that the coefficients are not constant, but the terms containing derivatives are all of the form $$x^k \frac{d^k}{dx^k} V,$$ which means that substituting $y=\log{x}$, $$\frac{d}{dy} = \frac{1}{x} \frac{d}{dx}, \quad \text{or} \quad \frac{d}{dx} = e^y \frac{d}{dy}$$ which allows you to reduce it to a constant-coefficient equation (but beware that derivatives higher than the first create more than one term when you use the chain rule).
• Solving the constant-coefficient equation you generate gives a function that is a sum of terms of the form $e^{\lambda y}$. Substituting $x$ back in, $e^{\lambda y} = e^{\lambda \log{x}} = x^{\lambda}$. Oct 11, 2015 at 12:13
your solution should be $Y=Cx^{m}$ not $Y=Ce^{mx}$