The Method of Separation of Variables

Separation of variables is simply a guess that the solution is a fixed shape that just gets turned up or down over time. The name says what it does: it pulls the $x$ part and the $t$ part apart into separate functions, each depending on one variable only. It's the standard first tool for heat, vibration and quantum problems, because it turns one hard PDE into two ODEs you already know how to solve, and then we add up lots of those simple solutions to build the real one. $u = X(x)\,T(t)$ looks fancy, but it's two pieces: $X(x)$ (the shape along the rod) and $T(t)$ (a dial that sets how big that shape is at each moment). There are a few steps after that, but each one is familiar, so bear with it.

When it works: the PDE and the boundary conditions must be linear and homogeneous. Linearity is what lets us add solutions together at the end, and zero boundary values are what let each piece satisfy them on its own. (The next lessons show how to handle nonzero boundaries.)

The steps, for $u_t = k\,u_{xx}$ with $u(0, t) = u(L, t) = 0$:

1. Substitute $u = X(x)T(t)$: $XT' = kX''T$.
2. Separate: $\frac{T'}{kT} = \frac{X''}{X}$. A function of $t$ equals a function of $x$ only if both are a constant, $-\lambda$.
3. Solve the eigenvalue problem $X'' + \lambda X = 0$, $X(0) = X(L) = 0$: nonzero solutions exist only for $\lambda_n = \left(\frac{n\pi}{L}\right)^2$, with $X_n = \sin\frac{n\pi x}{L}$.
4. Solve $T' = -k\lambda_nT$: $T_n = e^{-k\lambda_nt}$.
5. Superpose: $u = \sum B_nX_nT_n$, and choose $B_n$ so that $u(x, 0) = f(x)$: a Fourier sine series.

Why step 2 gives a constant. The left side only knows about $t$, and the right side only knows about $x$. Hold $t$ fixed and slide $x$ around: the left side can't change, so the right side can't either. Hold $x$ fixed and the same argument works the other way. So both sides equal one number, which we call $-\lambda$ (the minus sign just makes the interesting values come out positive).

Where the boundary conditions come in. Notice that $X'' + \lambda X = 0$ is the same equation as an undamped spring from ODEs. On its own it has solutions for every $\lambda$, but we also need $X$ to be zero at both ends, and that's a much stronger demand. A sine fits only if a whole number of half-waves spans the rod, which is exactly $\lambda_n = \left(\frac{n\pi}{L}\right)^2$. The last step is then the Fourier sine series from the previous unit.

A common mistake. It's tempting to think any $\lambda$ will do, since $X'' + \lambda X = 0$ can be solved for every $\lambda$. It can, but almost none of those solutions are zero at both ends. For $\lambda \le 0$ the solutions are straight lines or exponentials, and the only one with $X(0) = X(L) = 0$ is $X \equiv 0$. For $\lambda > 0$, only the special values $\lambda_n$ work. That's why they get a name: eigenvalues ("eigen" is German for "own," so these are the problem's own special values).

Separation of variables is the standard first attack on heat flow, vibration, electrostatics and quantum mechanics (the time-independent Schrödinger equation comes from exactly this split).