The Euler-Poisson equation

Question

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=\sinh(\pi)$$ 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? ${{{}}}$

Answer 1

$$x^{(4)}=-4x$$ $$x(t)=C_1 e^{-t} sin(t)+C_2 e^{t} sin(t)+C_3 e^{-t} cos(t)+C_4 e^{t} cos(t)$$ $$C_1=1/2; C_2=-1/2; C_3=C_4=0$$ $$x(t)=-sin(t) sinh(t)$$