Separation of variables

A method for solving differential equations by expressing a solution as a product of functions, each depending on only one independent variable.

Separation of variables for ordinary differential equations: A technique for solving an ordinary differential equation by placing dependent-variable terms and independent-variable terms on opposite sides. This simpler form illustrates separation by rearranging rather than factoring a multivariable solution.
Heat equation: A partial differential equation describing the diffusion of heat or other quantities. Separation produces spatial eigenfunctions whose coefficients decay exponentially in time.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple independent variables. Separation of variables is principally a technique for solving this broader class of equations.
Method of characteristics: A technique that solves certain first-order partial differential equations along curves in the domain. It follows characteristic curves rather than factoring a solution into variable-specific functions.
Product ansatz: A trial solution written as a product of functions, often with each function depending on a different variable. The product ansatz is the key assumption that makes variable-by-variable separation possible.
Wave equation: A partial differential equation describing the propagation of waves through space and time. Separating space from time reduces many wave problems to ordinary differential equations.
Ordinary differential equation: An equation involving an unknown function of one variable and its ordinary derivatives. Each separated factor commonly satisfies an ordinary differential equation.
Fourier transform: An integral transform that expresses a function in terms of its frequency components. Transforms can reduce differential equations to algebraic or lower-dimensional equations without a product ansatz.
Separation constant: A constant introduced when expressions depending on independent variables are set equal across a domain. Equating separated expressions forces each variable-dependent side to equal a constant.
Laplace's equation: The equation stating that a function's Laplacian is zero, describing harmonic fields in source-free regions. Its boundary-value problems in simple geometries often admit separated harmonic solutions.

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