Solución:
\(v=C_1\ sen\ Ax+G\cos Ax\)
\(v=\frac{dv}{dx}C_1\ A\ \cos\ Ax\ -G\ A\ sen\ Ax\)
\(v=\frac{d^2v}{dx^2}=-C_{1\ }A^{2\ }5m\ X_x-6x^2\cos X_X\)
\(-C_1\ x^2\ sen\ Ax\ -\ G\ x^{2\ }\cos\ Ax+\left(\frac{P}{EI}\right)\left(C_1\ sen\ Ax+G\ \cos\ Ax\ \right)=0\)
\(-C_1\ x^2\ sen\ Ax\ -\ G\ x^{2\ }\cos\ Ax\ .\ C_1\left(\frac{P}{EI}\right)sen\ Ax\ +\ G\ \left(\frac{P}{EI}\right)\cos\ Ax\ =0\)
\(C_1\sin\ Ax\ \left(\frac{P}{EI}-x^n\right)+C_2\ \cos\ 2\ Ax\ \left(\frac{P}{EI}-x^2\right)=0\)
\(\frac{P}{EL}=A^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A=\sqrt{\frac{P}{EL}}\)
\(v=C_1\ sen\ \sqrt{\frac{P}{EI}}+C_2\ \cos\ \sqrt{\frac{P}{EI}}\)
\(a\left(b+c\right)\) \(a=ab+ac\)
Calculando los valores de las constantes C1 y C2
\(v=0\ \ I\ \ x=0\)
\(v=0\ \ I\ \ x=L\)
\(x=0\)
\(v=0\)
\(C_1\ sen\ \sqrt{\frac{P}{EI}}\cos\ +\ C_2\ \cos\ \sqrt{\frac{P}{EI}}\left(0\right)=0\)
Para \(v=0\ \ \ \ \ I\ \ \ \ x=L\)
\(v\left(x=L\right)=C_1\ \sin\ \sqrt{\frac{P}{EI}}L=0\)
\(\sin\left(\sqrt{\frac{P}{EL}L}\right)=0\ \ \)
\(\sqrt{\frac{P}{EI}}L=n\pi\)
\(Pu=\frac{\pi^2EI}{L^2}\)
\(P=\frac{n^2\pi^2EI}{L^2}\)