How do I solve:
$$u_t = -u_{xxxx} + \pi^2u_{xx},$$ with BCs: $u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0$ and initial condition $u(x,0)=\cos(\pi x)$.
We have been told that we can use separation of variables however I can't seem to get the required solution $\cos(\pi x)\exp(-2\pi^4t)$.
Hint: Assume $u=X(x)T(t)$, you will get
$$XT'=-X^{(4)}T+\pi^2X''T=T(-X^{(4)}+\pi^2X'')$$ $$\frac{X}{-X^{(4)}+\pi^2X''}=\frac{T}{T'}$$
Then you can continue from there.