5.6 Using the Method of Variation of Parameters for the Forced Response

In certain nonhomogeneous ODEs, the right side f(t) cannot be determined by the method of undetermined coefficients. For these ODEs we must use the method of variation of parameters. This method will work with all linear equations including those with variable coefficients such as

provided that the general form of the natural response is known.

Our discussion will be restricted to second order ODEs with constant coefficients.

The method of variation of parameters replaces the constants k ₁ and k ₂ by two variables u ₁ and u ₂ that satisfy the following three relations:

Simultaneous solution of (5.68) and (5.69) will yield the values of du ₁/dt and du ₂/dt; then, integration of these will produce u ₁ and u ₂, which when substituted into (5.67) will yield the total solution.

Example 5.12

Find the total solution of

in terms of the constants k ₁ and k ₂ by the

1. method of undetermined coefficients

2. method of variation of parameters

Solution:

With either method, we must first find the natural response. The characteristic equation yields the roots s ₁ = ?1 and s ₂ = ?3. Therefore, the natural response is

1. Using the method of undetermined coefficients we let y _F = k ₃ (a constant). Then, by substitution into (5.71) we obtain k ₃ =...