# How Do I Find the Exact solution after using the Finite Difference Method?

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.

• Your expressions are hard to read. Please use MathJax. As for the question (if I understand it correctly), in general an approximation doesn't help you find the exact solution. Hence, it is used when no exact solution is available. Otherwise, why even use a numerical method? – Yuriy S Nov 23 '18 at 0:29
• Yeah I'm trying to use numerical methods. I don't know which one to use. – Jackie Nov 23 '18 at 0:29
• Your title says "finite differences", I assume this is the method you wanted to use? Honestly, I'm not sure what you are asking – Yuriy S Nov 23 '18 at 0:31
• This method is what I'm referring to: mathforcollege.com/nm/mws/gen/08ode/… I cant understand what he does on page five to solve for his homogeneous solution – Jackie Nov 23 '18 at 0:32
• Are you trying to solve the equation (E.1.1)? Cause that is not the equation you have written in this question. You missed a term – Yuriy S Nov 23 '18 at 0:39

## 1 Answer

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.

• Then what is this guy: mathforcollege.com/nm/mws/gen/08ode/… doing on page 5 after getting his matrix of approximated values at those nodes. – Jackie Nov 23 '18 at 0:50
• It appears he is using a completely different technique in order to derive the “exact” solution to the differential equation, in order to compare the relative accuracy of the finite difference method. The point is that the finite difference method is unable to provide exact solutions, and is meant primarily for cases where exact solutions are exceedingly difficult or even impossible to derive. – silvascientist Nov 23 '18 at 1:06