Variation of Parameters

A general particular-solution method for second-order linear equations with variable coefficients.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

For normalized $y''+p(x)y'+q(x)y=g(x)$ with fundamental homogeneous solutions $y_1,y_2$ and Wronskian $W=y_1y_2'-y_1'y_2\ne0$, variation of parameters seeks $y_p=u_1y_1+u_2y_2$ where $u_1'=-y_2g/W$ and $u_2'=y_1g/W$.

Notation and mathematical language

The auxiliary condition $u_1'y_1+u_2'y_2=0$ eliminates second derivatives of the unknown parameter functions. If the original leading coefficient is $a(x)$, divide first; otherwise the $g$ in the formulas is wrong. Constants arising in $u_1,u_2$ can be set to zero because they add homogeneous terms.

Conceptual picture

The method lets the coefficients of homogeneous modes vary just enough to absorb forcing. The Wronskian denominator solves a two-by-two system for $u_1',u_2'$, so its nonzero condition is the same independence requirement needed for a fundamental set.

Fully worked example

Interpretation and application

Variation of parameters handles forcing outside the undetermined-coefficients catalogue and variable coefficients when a homogeneous basis is known. Exact integral expressions may be more informative than unstable decimal antiderivatives and should be distinguished from numerical approximations.