## Variation of Parameters

For a second-order Ordinary Differential Equation,

 (1)

Assume that linearly independent solutions and are known. Find and such that
 (2) (3)

Now, impose the additional condition that
 (4)

so that
 (5) (6)

Plug , , and back into the original equation to obtain
 (7)

 (8)

Therefore,
 (9) (10)

Generalizing to an th degree ODE, let , ..., be the solutions to the homogeneous ODE and let , ..., be chosen such that

 (11)

Then the particular solution is
 (12)

© 1996-9 Eric W. Weisstein
1999-05-26