Real or complex numbers for the general solution of $y''(x)+k^2y(x)=0$?

Question

For $y''(x)+k^2y(x)=0$, we have the general solution $$ y(x)=Ae^{-ikx}+Be^{ikx} $$ But are the constants $A$ and $B$ real or complex numbers?

Answer 1

That depends on the initial conditions. In general these constants will be complex. For real initial conditions you get a completely real solution of the ODE, thus $y(x)=\overline{y(x)}$. This implies the additional condition $B=\bar A$ on the complex coefficients. Then the solution is $$y(x)=2\,{\rm Re}(\bar A e^{ikx})=C\cos(kx)+D\sin(kx)$$ where $C=2\,{\rm Re}(A)$ and $D=2\,{\rm Im}(A)$ are real constants.