Separation of Variables

PDEs · Axiom Academy

LESSON Separation of Variables A systematic method for solving the wave equation by splitting it into simpler ordinary differential equations We begin by assuming the solution can be written as a product of two functions: one depending only on position, the other only on time. 2. Substituting into the Wave Equation We substitute u(x,t) = X(x)T(t) into the wave equation. When we differentiate the product, we get partial derivatives that act on each function separately. The wave equation is u_tt = c²u_xx. After substitution and using the product rule: Now we divide both sides by X(x)T(t) to separate the variables. The left side depends only on t, while the right side depends only on x. 4. Obtaining Two Ordinary Differential Equations Setting each side equal to -λ gives us two separate ordinary differential equations, one for X(x) and one for T(t). 5. Boundary Conditions Determine Eigenvalues The boundary conditions on the spatial domain determine which values of λ (eigenvalues) are allowed. For example, if u(0,t) = u(L,t) = 0 (fixed endpoints), then X(0) = X(L) = 0. 6. Combining Solutions via Superposition The general solution is formed by taking a linear combination (superposition) of all the separated solutions, one for each eigenvalue.

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