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Particular Solution of 4th degree differential equation (column buckling)

I'm trying to use Mathematica to solve the Euler Column buckling problem (, but its not working out. Mathematica only shows me the zero solution. Is there any way of getting the no zero displacements solutions? For now, what I have is basically this:

equation = EIy''[x] + Py[x] == 0

DSolve[{equation, y[0] == 0, y[L] == 0}, y[x], x]

Out[143]= {{y[x] -> 0}}

Thanks, Luís Valarinho

POSTED BY: Luis Valarinho
6 Replies

Replace y[L] ==0 with y'[0] == yp0 and then combine y[L] ==0 with the solution you get.

POSTED BY: Frank Kampas

The solution given is correct but it isn't the only solution. Yes, you'll have to handle the boundary condition at L separately.

POSTED BY: Frank Kampas

I recommend not using capital letters for your constants (or at least give them unique names) because for example I is the symbol for Sqrt[-1] and E is the euler number Here is how I would solve it:

In[13]:= Clear["Global`*"]

In[14]:= sol = 
 DSolve[{y''[x] + ps/(es is) y[x] == 0, y[0] == 0}, y[x], x]

Out[14]= {{y[x] -> C[2] Sin[(Sqrt[ps] x)/(Sqrt[es] Sqrt[is])]}}

In[15]:= Solve[Sqrt[ps] l/Sqrt[es is] == n Pi, ps]

Out[15]= {{ps -> (es is n^2 \[Pi]^2)/l^2}}
POSTED BY: Kay Herbert

So, if I understood Mathematica doesn't solve the equation automatically.

The last boundary condition will have to be me adding it, right?

POSTED BY: Luis Valarinho

Ok, thanks!

POSTED BY: Luis Valarinho

Hello Kay!

Thanks for the suggestion. What I did actually was using the nomenclature that we use in Civil Engineering But you're right, it can lead to mistakes. Thanks for the other suggestions too. I was actually wondering if it is possible to do it in one step. We cannot ask Mathematica to give all the possible solutions? Or adding an assumption that C[2] should be different than zero? (something that is not necessarily truth).

Thanks, Luís Valarinho

POSTED BY: Luis Valarinho
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