Most equations don't let you cheat like this. If you know two solutions of a typical nonlinear equation, knowing their sum buys you nothing — you have to start the whole solving process over. But for an enormous and important family of PDEs, there is a genuine shortcut: once you have found any two solutions, you already know infinitely many more, for free, with zero extra work.

This is the single most useful consequence of linearity, and it is the engine behind almost every technique for solving PDEs by hand — it's exactly why Fourier series turn out to be the right tool for solving the heat and wave equations, rather than just a curiosity from a different branch of maths.

Worked example: checking it on the heat equation

Let's not just trust the one-line proof — let's watch it happen on a real PDE. The heat equation is u_t = \alpha\,u_{xx}, which is linear and homogeneous (every term has u or a derivative of u to the first power, and there's no term free of u).

Suppose u_1(x,t) = \sin(x)\,e^{-\alpha t} and u_2(x,t) = \sin(3x)\,e^{-9\alpha t}. A quick check shows each one alone solves the heat equation: for u_1, (u_1)_t = -\alpha\sin(x)e^{-\alpha t} and (u_1)_{xx} = -\sin(x)e^{-\alpha t}, so (u_1)_t = \alpha (u_1)_{xx}. ✓ The same pattern works for u_2 (the exponent 9\alpha exactly matches the second x-derivative of \sin(3x), which brings down a factor of 9).

Now form u = u_1 + u_2 and plug it directly into the equation, using nothing but the fact that derivatives are linear — the derivative of a sum is the sum of the derivatives:

There it is — u = u_1 + u_2 satisfies the heat equation too, and we never had to compute a single new derivative from scratch. We just reused the two pieces we already had, because both differentiation and the heat equation are linear operations end to end.

Why this is the whole strategy

Superposition doesn't stop at two terms — it extends to infinite sums (with mild convergence care), and that is exactly what makes the coming methods work. The plan is always the same three steps: find a stockpile of simple "building-block" solutions — products like e^{-\lambda t}\sin(kx), one for every allowed frequency k — then add them together in just the right amounts to match whatever starting shape or boundary condition you were actually given. The "right amounts" are the Fourier coefficients, and the technique for finding the building blocks in the first place is called separation of variables — the next big idea in this course.

This is far more than a piece of paperwork. A guitar string doesn't just wobble in one shape — its motion is a superposition of a fundamental mode plus a whole ladder of overtones, each one on its own a perfectly valid solution of the wave equation, all added together to make the actual sound you hear. A skyscraper swaying in an earthquake is doing the same thing: engineers decompose the building's response into a handful of simple normal modes of vibration, work out how each one behaves on its own (easy), and then superpose them to predict the real, complicated motion (otherwise nearly impossible). Wherever you see a complicated linear system described as a "sum of simple modes," superposition is the reason that description is allowed to work at all.

None of that plan survives contact with a nonlinear PDE. You cannot add two solutions of Burgers' equation u_t + u\,u_x = 0 and expect a third solution to fall out — the u\,u_x term mixes u_1 and u_2 together in a way that simply doesn't cancel. That single difference is why nonlinear PDEs are so much harder to solve by hand, and why this course spends most of its time in the linear world, where addition is always allowed.

Add two modes

Two building-block solutions on [0, \pi], u_1 = \sin x and u_2 = \sin 3x (faint), and their superposition c_1 u_1 + c_2 u_2 (bold). Dial the weights: every combination is again a valid solution, and by choosing the weights you sculpt the shape. This is a Fourier series in miniature.