The PDE $$k^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ is separable, so we'll look for solutions of the form $$u(x,t) = X(x)T(t).$$ This leads us to a Sturm-Liouville problem.

Plug the formula for $u$ into the PDE, then separate the variables.
Note: In your answer, put all the $X$'s on the left answer blank. Put all the $T$'s and the constant $k$ in the right answer blank. Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\prime$. Do NOT use $X^\prime(x)$
The result is
= = $-L$
where $L$ is a constant.

This separates into two independent ordinary differential equations, which can be solved separately
DE in X: $= 0$
DE in T: $= 0$

Find the general solutions for the these these ODEs, and use them to find the solution $u(x,t)$.
Use $A$ and $B$ for arbitrary constants in the general solution for $X$.
Use $C$ and $D$ for arbitary constants in the general solution for $T$.
There are three cases:
Case 1: $L = 0$
$X(x) =$ (A,B are constants.)
$T(t) =$ (C,D are constants.)
$u =$
Case 2: $L = -w^2$ where $w$ is some positive number.
$X(x) =$ (A,B are constants.)
$T(t) =$ (C,D are constants.)
$u =$
Case 3: $L = w^2$ where $w$ is some positive number.
$X(x) =$ (A,B are constants.)
$T(t) =$ (C,D are constants.)
$u =$