How do you find a Power Series solution of a nonhomogeneous differential equation?

Question

Emilia Aguilar · Accepted Answer

Assuming you know how to find a power series solution for a linear differential equation around the point #x_0#, you just have to expand the source term into a Taylor series around #x_0# and proceed as usual. This may add considerable effort to the solution and if the power series solution can be identified as an elementary function, it's generally easier to just solve the homogeneous equation and use either the method of undetermined coefficients or the method of variation of parameters.

Isabella Ackerman · Answer

undefined To find a power series solution of a nonhomogeneous differential equation, follow these steps:

1. Solve the homogeneous part of the equation to find the complementary solution.
2. Assume the particular solution has a power series representation and substitute it into the nonhomogeneous equation.
3. Equate coefficients of like powers of the variable in the equation.
4. Solve for the coefficients of the power series by solving the resulting recurrence relations.
5. Combine the complementary solution with the particular solution to get the general solution of the nonhomogeneous equation.