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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.

silvascientist
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Jackie
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    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 00:29
  • Yeah I'm trying to use numerical methods. I don't know which one to use. – Jackie Nov 23 '18 at 00:29
  • 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 00:31
  • This method is what I'm referring to: http://mathforcollege.com/nm/mws/gen/08ode/mws_gen_ode_spe_finitedif.pdf I cant understand what he does on page five to solve for his homogeneous solution – Jackie Nov 23 '18 at 00:32
  • 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 00:39
  • But if you really want to solve the equation you written here, then the homogeneous solution is the general solution of $y''=0$ which is just a linear function $Ax+B$ – Yuriy S Nov 23 '18 at 00:42
  • I really have no idea how to solve that. Just like in that document, I made my matrix, got my values. how would I set up my homogeneous and particular equations in order to follow the next steps? – Jackie Nov 23 '18 at 00:48
  • Jackie, finding an exact solution is certainly possible here (and I just hinted you the homogeneous part), and it's also possible to get a numerical solution. But those two things are not related to each other. Aside from originating from the same equation – Yuriy S Nov 23 '18 at 00:50
  • Right..Ax+B. Doesn't that mean there are infinite homogeneous solutions? How Do I know what to pick. Does it matter? – Jackie Nov 23 '18 at 00:52
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    You usually have initial or boundary conditions... Which are also necessary to get any sensible numerical solution – Yuriy S Nov 23 '18 at 00:54
  • So if I have i(1) = 0, i(2) = -1.47, i(3) = -1.47, and i(4) = 0. How Do I use that? I really need guidance here. – Jackie Nov 23 '18 at 01:12
  • What is $i$? I still don't understand what you are trying to do, and I can't study the PDF you linked in detail right now. Maybe you could clarify what and why are you trying to do and I will come back in a few hours – Yuriy S Nov 23 '18 at 01:15
  • I'm trying to approximate the deflection of the beam, with its moment function described by the differential equation: M = w/2(L-x)^2. So, y'' = w/2(L-x)^2. I then substitute y'' for the central divided difference formula in order to approximate y(2) and y(3). Beam mechanics has it so that the deflection of beams are 0 at the supports --> y(1) and y(4) = 0. So I sub the CDD form and the values of w, E, I, and L. Make a linear syste – Jackie Nov 23 '18 at 01:19
  • Then I solve the linear system for my solutions: y(1) = 0 y(2) = -1.47 y(3) = -1.47 y(4) = 0 I am now trying to compare this approximated value to the exact value. But I don't know how to derive the equation for this exact value. (Compare by relative true error btw) – Jackie Nov 23 '18 at 01:21

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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.

silvascientist
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  • Then what is this guy: http://mathforcollege.com/nm/mws/gen/08ode/mws_gen_ode_spe_finitedif.pdf

    doing on page 5 after getting his matrix of approximated values at those nodes.

     – Jackie Nov 23 '18 at 00:50
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    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 01:06