Why is the solution for a second order ODE -m_{l}^{2} e^{im_{l}\phi}?

[kasse]

Can someone explain to me why the solution of $$\frac{d^{2}\Phi (\phi)}{d\phi^{2}} = -m_{l}^{2}$$ is $$\Phi = e^{im_{l}\phi}$$?

 

[Dick]

It's not. A solution of
$$\frac{d^{2}\Phi (\phi)}{d\phi^{2}} = -m_{l}^{2}\Phi(\phi)$$ is $$\Phi = e^{im_{l}\phi}$$.
Just substitute Phi into the ODE.

 

[kasse]

If I'm given the ODE, is inspection the only way to find the solution?

 

[HallsofIvy]

If you have not learned to solve differential equations, yes!

If you have then you would know how to use the solutions to the characteristic equation, then you could use that method.