Linear Differential Equation - Nonhomogeneous Equation With Constant Coefficients

Nonhomogeneous Equation With Constant Coefficients

To obtain the solution to the nonhomogeneous equation (sometimes called inhomogeneous equation), find a particular integral y_P(x) by either the method of undetermined coefficients or the method of variation of parameters; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular integral. Or, when the initial conditions are set, use Laplace transform to obtain the particular solution directly.

Suppose we face

For later convenience, define the characteristic polynomial

We find the solution basis as in the homogeneous (f(x)=0) case. We now seek a particular integral y_p(x) by the variation of parameters method. Let the coefficients of the linear combination be functions of x:

For ease of notation we will drop the dependency on x (i.e. the various (x)). Using the operator notation D = d/dx, the ODE in question is P(D)y = f; so

With the constraints

the parameters commute out,

But P(D)y_j = 0, therefore

This, with the constraints, gives a linear system in the u′_j. This much can always be solved; in fact, combining Cramer's rule with the Wronskian,

The rest is a matter of integrating u′_j.

The particular integral is not unique; also satisfies the ODE for any set of constants c_j.