Verify that the differential equation:

$\displaystyle \frac{d^2y}{dx^2} = -k^2y$

has as its solution:

$\displaystyle y = A\cos{(kx)} + B\sin{(kx)}$

where A and B are arbitrary constants. Show also that this solution can be written in the form:

$\displaystyle y = C\cos{(kx+\alpha)} = C \ Re[e^{j(kx + \alpha)}] = Re[Ce^{j\alpha}e^{jkx}]$

And express C and $\displaystyle \alpha$ as functions of A and B.