Using variation of parameters to solve a system of nonhomogeneous differential equations

If undetermined coefficients isn’t a viable method for solving a nonhomogeneous system of differential equations, we can always use the method of variation of parameters instead. Just like with undetermined coefficients, we have to start by finding the corresponding complementary solution, which is the general solution of the associated homogeneous equation.

Using variation of parameters with a system of equations to find the particular solution

Like the method of undetermined coefficients, variation of parameters is a method you can use to find the general solution to a second-order (or higher-order) nonhomogeneous differential equation. Remember that homogenous differential equations have a 0 on the right side, where nonhomogeneous differential equations have a non-zero function on the right side.

Solution to a nonhomogeneous equation using variation of parameters

Like the method of undetermined coefficients, variation of parameters is a method we can use to find the general solution to a second-order (or higher-order) nonhomogeneous differential equation. We’ll look at variation of parameters, as well as how to use Cramer’s rule with variation of parameters.

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