I want to find the stability of $$u_t + (1 + \pi^2)u_x + u_{xxx} = 0$$ Applying forward euler method with forward and central differenve schemes, I get $$ \frac{U_n^{j+1} - U_n^j}{k} + (1 + \pi^2)\frac{U_{n+1}^j - U_n^j}{h} + \frac{U_{n+3}^j - 3U_{n+1}^j + 3U_{n-1}^j - U_{n-3}^j}{8h^3} $$ Substituting $$ U_j^n = \xi^n\cdot \exp \left (2\pi i \beta \frac{j}{M+1} \right) $$ such that the method is stable for $|\xi| \leq 1$, I get a 6-th degree polynomial which seems impossible to solve.
Am I having the right approach with some wrong assumptions or should the problem be solved using another aproach?