For $k > 0$, the PDE $$k\frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial t}$$ is separable, so we follow the Sturm-Liouville procedure and look for basic solutions of the form $u(x,t) = X(x)T(t)$
Using this basic solution the PDE can be rewritten, collecting $X$ and its derivatives on the left and $k$ and $T$ and its derivatives on the right, as
= = $-\lambda$

Note: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\prime$. Do NOT use $X^\prime(x)$

Since these differential equations are independent of each other, they can be separated: (Type the word lambda to represent the Greek letter $\lambda$.)
DE in X: $= 0$

DE in T: $= 0$

In each case the general solution in $X$ is written with constants $a$ and $b$ and the general solution in $T$ is written with constant $c$. (Use the word alpha to represent the Greek letter $\alpha$.)
Case 1: $\lambda = 0$
$X(x) =$ with $X(0)=a$ and $X'(0)=b$.
$T(t) =$ with $T(0)=c$.
$u =$
Case 2: $\lambda = -\alpha^2$
$X(x) =$ with $X(0)=a$ and $X'(0)=\alpha b$.
$T(t) =$ with $T(0)=c$.
$u =$
Case 3: $\lambda = \alpha^2$
$X(x) =$ with $X(0)=a$ and $X'(0)=\alpha b$.
$T(t) =$ with $T(0)=c$.
$u =$

In each case solutions to the original PDE are obtained by summing the basic solutions, choosing the constants to make the sum satisfy boundary conditions.