Variation of Parameters

(1)

Assume that linearly independent solutions and are known to the homogeneous equation

(2)

and seek and such that

			(3)
			(4)

Now, impose the additional condition that

(5)

so that

			(6)
			(7)

Plug , , and back into the original equation to obtain

(8)

which simplifies to

(9)

Combing equations (◇) and (9) and simultaneously solving for and then gives

			(10)
			(11)

where

(12)

is the Wronskian, which is a function of only, so these can be integrated directly to obtain

			(13)
			(14)

which can be plugged in to give the particular solution

(15)

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

${y_1v_1^'+y_2v_2^'+...+y_nv_n^'=0; y_1^'v_1^'+y_2^'v_2^'+...+y_n^'v_n^'=0; |; y_1^((n-1))v_1^'+y_2^((n-1))v_2^'+...+y_n^((n-1))v_n^'=g(x).$

(16)

and the particular solution is then

(17)

See also