Method of Variation of Parameters

The method of variation of parameters is a general method for concluding solutions to nonhomogenous differential equations. If the differential equation at hand satisfies requirements of the method of undetermined coefficients, the latter is usually more efficient.

Workflow

Given a nonhomogenous differential equation $a{y}^{\prime \prime }+b{y}^{\prime }+cy=G(x)$ the associated homogenous differential equation is to be solved for its solutions $y\_1\text{and}y\_2$. (for example using the auxiliary equation)

We then construct a system of equations with unknown functions $v\_1^{\prime }\text{and}v\_2^{\prime }$

$$\{\begin{array}{l}v\_1^{\prime }y\_1+v\_2^{\prime }y\_2=0\\ v\_1^{\prime }y\_1^{\prime }+v\_2^{\prime }y\_2^{\prime }=\frac{G(x)}{a}\end{array}$$

where $a$ is taken from the initial equation $a{y}^{\prime \prime }+b{y}^{\prime }+cy=G(x)$. This system can be solved using Cramer’s Rule giving $v\_1^{\prime }\text{and}v\_2^{\prime }$, which have to be integrated to get

$$v\_1=\int v\_1^{\prime }dx\text{and}v\_2=\int v\_2^{\prime }dx$$

The particular solution $y\_p$ is then given by

$$y\_p=v\_1y\_1+v\_2y\_2$$

and the general one $y$ with

$$y=y\_c+y\_p$$

where $y\_c$ is the general solution to the differential equations homogenous associated differential equation.