Method of Variation of Parameters

Proof Technique
The method of variation of parameters is a technique for finding a particular solution to a nonhomogeneous linear second order ODE:
 * $(1): \quad y'' + P \left({x}\right) y' + Q \left({x}\right) y = R \left({x}\right)$

provided that the general solution of the corresponding homogeneous linear second order ODE:
 * $(2): \quad y'' + P \left({x}\right) y' + Q \left({x}\right) y = 0$

is already known.

Method
Let the general solution of $(2)$ be:
 * $y = C_1 y_1 \left({x}\right) + C_1 y_2 \left({x}\right)$

Then a particular solution of $(1)$ is:
 * $\displaystyle y = y_1 \int -\frac {y_2 \left({x}\right) R \left({x}\right)} {W \left({y_1, y_2}\right)} \, \mathrm d x + y_2 \int \frac {y_1 \left({x}\right) R \left({x}\right)} {W \left({y_1, y_2}\right)} \, \mathrm d x$

where $W \left({y_1, y_2}\right)$ denotes the Wronskian of $y_1 \left({x}\right)$ and $y_2 \left({x}\right)$.

Proof
Let the general solution of $(2)$ be:
 * $(3): \quad y = C_1 y_1 \left({x}\right) + C_1 y_2 \left({x}\right)$

Let the arbitrary constants $C_1$ and $C_2$ be replaced by functions $v_1 \left({x}\right)$ and $v_2 \left({x}\right)$.

It is required that $v_1$ and $v_2$ be determined so as to make:
 * $(4): \quad y = v_1 \left({x}\right) y_1 \left({x}\right) + v_2 \left({x}\right) y_2 \left({x}\right)$

a particular solution of $(1)$.

Then:

Suppose ${v_1}' y_1 + {v_2}' y_2$ were made to vanish:
 * $(6): \quad {v_1}' y_1 + {v_2}' y_2 = 0$

Then:

Hence:

Because $y_1$ and $y_2$ are both particular solutions of $(2)$:
 * ${y_1} + P {y_1}' + Q y_1 = {y_2} + P {y_2}' + Q y_2 = 0$

and so from $(9)$:
 * $(10): \quad {v_1}' {y_1}' + {v_2}' {y_2}' = R \left({x}\right)$

In summary:

We started with the assumption that:
 * $(3): \quad y = C_1 y_1 \left({x}\right) + C_1 y_2 \left({x}\right)$

and so $y_1$ and $y_2$ are linearly independent.

Thus by Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent:
 * $W \left({y_1, y_2}\right) \ne 0$

and so $(11)$ is defined.

Thus:

and so as required:
 * $\displaystyle y = y_1 \int -\frac {y_2 \left({x}\right) R \left({x}\right)} {W \left({y_1, y_2}\right)} \, \mathrm d x + y_2 \int \frac {y_1 \left({x}\right) R \left({x}\right)} {W \left({y_1, y_2}\right)} \, \mathrm d x$

Source of Name
The name method of variation of parameters derives from the method of operation: the parameters $C_1$ and $C_2$ are made to vary by replacing them with the functions $v_1 \left({x}\right)$ and $v_2 \left({x}\right)$.