Step 1: Separation of variables

Assume

$$u(x,t) = X(x) T(t).$$ (3)

Substituting into the PDE gives

$$X(x) T'(t) = X''(x) T(t) \quad \Rightarrow \quad \frac{T'(t)}{T(t)} = \frac{X''(x)}{X(x)} = -\lambda,$$ (4)

where $\lambda>0$ is a separation constant.