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

Questions
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1

I’m trying to use Mathematica to solve the Euler Column buckling problem (Column Buckling), 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

2

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

3

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

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

4

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

5

Ok, thanks!

6

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}}
7

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