The formula for the critical load of a column was derived in 1757 by Leonhard Euler, the great Swiss mathematician. Euler’s analysis was based on the differential equation of the elastic curve:
Equilibrio del momento requiere que MC X 
\(\frac{dv^2}{dx\ ^2}=FV\)
\(\frac{dv}{dx}\)=\(\frac{dv}{E1}\)
\(\frac{dv}{dx^2}\)+\(\left(\frac{p}{E1}\right)v=0\)
\(\left(\frac{dv^2}{dx^2}\right)+\left(\frac{p}{E1}\right)V=0\)
Para resolver una ecuación  diferencial debemos proponer una solución  que la satisface.
\(V=c1senx\ ʎ\ x+c1ʎx\)            O   \(V=C1SEN\sqrt{\frac{P}{C1\ }}X+C2\ COS\sqrt{\frac{P}{E1}}X\) 
\(V'\ =\frac{DV}{DX}=\ C1\ ʎ\ COS\ ʎ\ X^{^{ }}SEN\ ʎX\)
\(V'=\frac{dv^2}{dx^2}=c1ʎ\ ^2senʎ\ x-c1ʎ\cosʎx\)
\(c1ʎ^{2\ }senʎx-c1\cosʎx+\left(\frac{p}{E1}\right)\left(C1\ senʎx+c1\cosʎx\right)=0\)
\(c1ʎ^2senʎx-c2ʎ^2\cosʎx+c1\left(\frac{p}{E1}\right)sen\ ʎx+c1\ \frac{p}{E1}\ \left(\cosʎx\right)=0\)
\(c1\ sen2x\left(\frac{p}{E1}-ʎ^2\right)+c2\cosʎx\ \left(\frac{p}{E1}-ʎ^2\right)=0\)
\(\frac{p}{E1}=ʎ^2=\sqrt{ʎ\ }=\sqrt{\frac{p}{E1}}\)
ecuaciones de frontera 
v=0  1 x=0
v=0 1 x=0   calculando las variables c1 y c2 
x=0
v=0
\(c1\ sen\sqrt{\frac{p}{E1}\left(0\right)}+c2\cos\sqrt{\frac{p}{E1}}\left(0\right)=o\)
=)c2=0
para v=0 1 x  -l 
\(\left(v=x=L\right)=C1\ sen\sqrt{\frac{p}{E1}}l=0\)
\(sen\left(\sqrt{\frac{p}{E1}L}\right)=0\)
\(\sqrt{\frac{P}{E1}}L=M\pi\)
\(\frac{P}{E1}L^2=\pi^2\ \ N^2\)
\(P=\frac{N^2\ \pi^2\ E1}{L^2}\)
para calcular la carga critica= N1 
\(Pc=\ \frac{\pi^2\ e1}{l^2}\)
la c1 representa que tanto se pandea la columna