My starting equation is $y'' = \frac{wx}{2EI}(L-x)$ [Beam Formula]
I got my approximations, but how do I use that to find the exact equation? I know that y = y(homogeneous) + y(particular).
But the homogeneous solution would come from $y'' = 0$. How do I even use that to find the homogeneous solution with my characteristic equation?
Also, I find that my particular solution is also zero. (Guessing that the answer to: $y'' = 0$ is $y1 = y2 = 0$.)
Help, thanks.
What you are asking for is impossible. Finite difference methods are a way to generate numerical approximations to the solution of an equation, no more, no less. Exact solutions, when they exist, require different techniques to derive.