Mr. Integral
Math notes/Differential Equations/Second-Order Linear Equations

Variation of Parameters — Cheat sheet

CHEAT SHEET   ·   FREE

Choose “Save as PDF” in the print dialog.
On this page

Core rule

yp=−y1∫y2gWdt+y2∫y1gWdt.y_p=-y_1\int\frac{y_2g}{W}dt+y_2\int\frac{y_1g}{W}dt.

Watch for

Changing integration constants only adds homogeneous terms, so one convenient set is enough for a particular solution. Definite integrals from an initial time are useful for enforcing zero initial response and for numerical evaluation.