$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$
I know this is a ODE so this is what I came up with so far: $$i\omega^3\hat{U} -(-i)\frac{\partial}{\partial{\omega}}\alpha\hat{U} = 0$$ by canceling the the $i$ we are left with $$\omega^3\hat{U} +\frac{\partial}{\partial{\omega}}\alpha\hat{U} = 0$$ so
$\hat{U}(\omega) =C(\omega)e^{\frac{\omega^3}{\alpha}}$ After we take the Fourier transform of the equation and we get
$$u(x)= \int^\infty_{-\infty} c(x)e^{\frac{\omega^3}{\alpha}}e^{-i\omega x}$$ then I got very stumped, please any input is most welcome. I know how to solve for Pde but got confused for an ODE.