The Euler-Poisson equation $$\int_{0}^\pi (x''^2+4x^2) dt$$
$$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=\sinh(\pi)$$
This is The Euler-Poisson equation, i found:
$$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial x'} +\frac {d^2}{d^2t} \frac{\partial f}{\partial x''}=0$$
$$2x''''=8x$$
$$x''''(t)=4x;$$
decide which way next?
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$$x^{(4)}=-4x$$
$$x(t)=C_1 e^{-t} sin(t)+C_2 e^{t} sin(t)+C_3 e^{-t} cos(t)+C_4 e^{t} cos(t)$$
$$C_1=1/2; C_2=-1/2; C_3=C_4=0$$
$$x(t)=-sin(t) sinh(t)$$
