A method to obtain a particular solution of a linear inhomogeneous ordinary differential equation by allowing the constants of the homogeneous general solution to become functions of the independent variable and solving for those functions subject to auxiliary conditions (often using the Wronskian). It applies to variable- or constant-coefficient linear ODEs.

Principle

Exploit linear superposition by replacing constant coefficients in the homogeneous solution by unknown functions, imposing constraints to eliminate derivatives of those functions where advantageous, then solving the resulting first-order system using integrals that involve the fundamental solutions and the forcing term divided by the Wronskian.

Demonstration

For a second-order equation with fundamental solutions y1,y2 and forcing f(t), a particular solution can be written y_p = -y1 ∫ (y2 f/W) dt + y2 ∫ (y1 f/W) dt where W is the Wronskian of y1,y2. The integrals yield explicit particular solutions when computable.

Misapplication

Applying variation of parameters without first verifying a fundamental set of linearly independent homogeneous solutions or in non-linear equations; failing to check the Wronskian for zeros on the interval leads to invalid integrals.

Consequence

A broadly applicable constructive method that yields particular solutions when fundamental solutions are known; works for variable coefficients and forcings outside the special families suited to undetermined coefficients.

Reversal

When the forcing has a simple admissible form and coefficients are constant use the method of undetermined coefficients for an algebraic route; when more structure is present, Green's functions or integral transforms may be alternative formulations.

Boundary

Requires knowledge of a fundamental system of homogeneous solutions and a nonzero Wronskian on the interval of interest; integrals may be intractable in closed form and lead to expressions defined only up to indefinite integrals.

Semantic Tension

‘Variation of parameters’ versus ‘method of undetermined coefficients’ — both find particular solutions but differ in generality and computational form (integral vs algebraic ansatz); they can overlap in domains where both apply.

Synthesis

Variation of parameters converts the problem of a particular solution into solving for time-varying coefficients multiplying known homogeneous solutions; by imposing auxiliary constraints and integrating combinations involving the Wronskian and forcing term it produces particular solutions applicable even for variable-coefficient linear ODEs.