2.6.4—Linear differential equations

For ay″+by′+cy=F(x), write y=yc+yp. The complementary function yc solves the homogeneous equation; a particular integral yp accounts for the forcing term.

Choose a trial form matching F(x): exponentials, polynomials, sines/cosines or their combinations. If the trial duplicates part of yc, multiply by enough powers of x to make it independent.

For y″+y=cos x, cos x and sin x already belong to yc, so a trial such as Ax sin x is required rather than A cos x+B sin x.

The complementary function alone solves only the zero-forcing equation, and a trial form that duplicates yc cannot determine a particular solution.