Variation of Parameters for Systems of Equations

Given an initial value problem of a nonhomogeneous linear system of differential equations in the form:

where is a coefficient matrix, one can use the variation of parameters method to find the solution. Much like in undetermined coefficients method, we see that the general solution takes the form:

where is the complementary solution and is the particular solution.

To solve this equation we first need to find the fundamental matrix, , of matrix . After which we can skip over many paragraphs of math in the text book to bring you the only equation that matters.

Where is the inverse fundamental matrix.

See Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 283 and
1-Sect 4.8 Method of Variation of Parameters for sources and more information.