Separation Method

Differential Equations · Axiom Academy

The complete six-step process that turns one hard PDE into a family of easy ODEs — and stitches the pieces back into a full solution. 1. The Complete Six-Step Process Separation of variables is not one trick — it is a procedure you run in order, every time. Watch the six stages light up as a pipeline: each stage's output feeds the next. Assume product form: u(x,t)=X(x)T(t) Substitute into the PDE: replace u with the product Separate variables: all t -terms on one side, all x -terms on the other Set equal to a constant: both sides must equal the same separation constant Solve the ODEs + apply the BCs: find the eigenvalues and eigenfunctions Sum the solutions: the general solution is a series of particular solutions 2. Steps 1–2: Assume the Product & Substitute Take the heat equation on a rod of length L held at zero temperature at both ends: , u(0,t)=u(L,t)=0 , u(x,0)=f(x) The key move: assume the solution is a product of a function of x alone and a function of t alone. Watch that product get built — a fixed spatial shape X(x) riding a shrinking or growing time-envelope T(t) . Substituted into u_t = k u_ xx 3. Steps 3–4: Separate & Set Equal to a Constant Divide the substituted equation by X(x)T(t) . Everything depending on t lands on one side; everything depending on x lands on the other:

This is the written version of the interactive lesson above. See the full Differential Equations course.