The two most basic differential equation used often in Quantum mechanics is given below.
1. $ \frac{d^2 \phi}{dx^2} + \alpha^2 \phi = 0$
solution:
$\phi = A e^{i \alpha x} + B e^{-i \alpha x}$
or,
$\phi = A \sin \alpha x + B \cos \alpha x$
as, $ e^{i \theta} = \cos \theta + i \sin \theta $
2. $ \frac{d^2 \phi}{dx^2} - \alpha^2 \phi = 0 $
solution: $\phi = A e^{\alpha x} + B e^{- \alpha x}$