Step 2: Solve the spatial part

We solve

$$X''(x) + \lambda X(x) = 0, \quad X(0) = X(L) = 0.$$ (5)

The solution is

$$X_n(x) = \sin\Big(\frac{n \pi x}{L}\Big), \quad \lambda_n = \Big(\frac{n\pi}{L}\Big)^2, \quad n = 1,2,3,\dots$$ (6)