Consider a partial differential equation with the following homogeneous boundary conditions and initial condition:
\[\begin{align} \mathbf{PDE.}\quad &u_t(x, t) = \alpha^2u_{xx}(x, t),\quad 0 < x < L,\quad t > 0 \\ \mathbf{BC.}\quad &u(0, t) = u(L, t) = 0 \\ \mathbf{IC.}\quad &u(x, 0) = f(x) \end{align}\]
A common technique used to solve differential equations in this course is to assume the solution can be written as
\[u(x, t) = X(x)T(t)\]
\[\begin{align} u_t &= \alpha^2 u_{xx} \\ X(x)T'(t) &= \alpha^2X''(x)T(t) \\ \frac{T'(t)}{\alpha^2 T(t)} &= \frac{X''(x)}{X(x)} \end{align}\]
We can observe the LHS is independent of \(x\) and the RHS is independent of \(t\), so this implies both sides must be constant. Thus,
\(T'(t) = \alpha^2\lambda T(t)\) and \(X''(x) = \lambda X(x)\). Assuming that \(u\) is a nontrivial solution, we obtain \(X(0) = X(L) = 0\) from our original boundary conditions.
We can show \(X(x) = 0\) is the only solution for the case of \(\lambda \geq 0\) by solving the characteristic equation and using our new boundary conditions. For the case of \(\lambda = -\mu^2 < 0\),
\[\begin{align} X(x) &= c_1\cos(\mu x) + c_2\sin(\mu x) \\ X(0) &= c_1 = 0 \\ X(L) &= c_2\sin(\mu L) = 0 \end{align}\]
This implies \(\mu L = \pi n\) for some integer \(n\). Thus, the nontrivial solutions for \(X(x)\) can be written as \(X_n(x) = \sin\left(\frac{\pi n}{L}x\right)\) up to some scalar multiples. The corresponding solution for \(T_n(t)\) for \(X_n(t)\) is \(T_n(t) = e^{-(\alpha \pi n / L)^2 t}\).
Hence, our general solution is the linear combination
\[\begin{align} u(x, t) &= \sum_{n=1}^{\infty} b_n e^{-(\alpha \pi n / L)^2 t}\sin\left(\frac{\pi n}{L}x\right) \\ u(x, 0) &= f(x) = \sum_{n=1}^{\infty} b_n\sin\left(\frac{\pi n}{L}x\right) \end{align}\]
The above may look like a contradiction, but rest assured, it is possible to find coefficients \(b_n\), such that the right hand series can pointwise converge to any arbitrary function \(f(x)\). This series is a special case of what’s known as the Fourier Series and is also the birthplace of what’s known as Harmonic Analysis.