2.6.2—Linear differential equations

The general solution of a linear ODE is complementary function (the homogeneous equation) plus a particular integral (one solution for the forcing term). Initial conditions determine the constants.

For constant coefficients, solve the auxiliary equation for the complementary function. Choose a particular form matching the forcing, multiplying by x when it duplicates a complementary term.

For y″−3y′+2y=e^x, the auxiliary roots are 1 and 2; because e^x duplicates a complementary term, try Axe^x for the particular solution.

The complementary function alone cannot satisfy a non-zero forcing term, and a guessed particular form must be adjusted when resonance occurs.