Verifying Solutions

Differential Equations · Axiom Academy

EXAMPLE Verifying Solutions to the Heat Equation Verify that a proposed solution satisfies a partial differential equation by computing and comparing partial derivatives. Verify that is a solution of the heat equation u_t = u_ xx . Excellent work! You've successfully verified a solution to the heat equation. Here's what we learned: Verification process: To verify a PDE solution, compute the required partial derivatives and check whether the equation is satisfied. Time derivative: Differentiating with respect to t treats as a constant and applies the chain rule, giving . Spatial derivatives: Differentiating with respect to x treats e^ -t as a constant. The first derivative gives ; the second gives . Equality check: Both u_t and u_ xx equal , confirming the solution satisfies the heat equation. Physical interpretation: This solution represents heat diffusion with exponential decay in time and a sinusoidal spatial distribution. This systematic verification approach works for any proposed PDE solution — always compute derivatives carefully, watching for which variable is held constant in each differentiation.

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