Variation of Parameters Method

Differential Equations · Axiom Academy

LESSON Variation of Parameters Method Derive u_1' and u_2' from a constraint and a Wronskian — a method that finds a particular solution for ANY forcing term g(x) . 1. Trade Constants for Functions Start from the homogeneous solution y_h = c_1y_1 + c_2y_2 . To satisfy the nonhomogeneous equation y'' + p(x)y' + q(x)y = g(x) , we let the constants vary: seek a particular solution of the form Two unknown functions replace two constants Differentiating y_p once brings in u_1'y_1 + u_2'y_2 terms we don't want in the mix — so we're free to impose a constraint that kills them: 2. One Constraint, One Substitution, Two Equations With u_1'y_1+u_2'y_2=0 imposed, differentiate y_p' again and substitute into the original ODE. Every q(x) and p(x) term cancels against the fact that y_1,y_2 already solve the homogeneous equation — leaving one clean new equation. Together with the constraint, that's a linear system for the unknowns u_1' and u_2' : 3. Cramer's Rule Gives the Formulas Solve the system by Cramer's rule: swap each column for the right-hand side , take that determinant, and divide by W . Both unknowns come out with W in the denominator and the other y in the numerator: 4. The Complete Algorithm, Worked Four steps turn any y'' + p(x)y' + q(x)y = g(x) into a solved particular solution — watch the algorithm run on a concrete equation, : Step 4 integrates using , and u_2'=1 directly.

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