I'm solving a quantum mechanics problem for the particle in a potential well, and the equation I have to solve is $\frac{d^2\psi}{dx^2}+k\psi=0$where $k=\frac{2mE}{\hbar^2}$ This seems easy enough to solve. It is a second order linear differential equation with constant coefficient of the form $a\psi''(x)+b\psi'(x)+c\psi(x)=0$, so I thought we were to use the characteristic equation $ar^2+br+c=0$and solve for roots $r_1$ and $r_2$. Doing that, I get $r^2+k=0$and therefore $r=\pm \sqrt{k}$ The general solution is given by $\psi(x)=\exp\left(\sqrt{\frac{2mE}{\hbar^2}}x\right)+\exp\left(-\sqrt{\frac{2mE}{\hbar^2}}x\right)$
However, when I refer to Griffiths' Introduction to Quantum Mechanics, he finds the general solution to be $\psi(x)=A\sin kx+B\cos kx$ Where have I gone wrong? Thanks.