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:
 * $(1): \quad y'' + \map P x y' + \map Q x y = \map R x$

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

is already known.

Method
Let the general solution of $(2)$ be:
 * $y = C_1 \, \map {y_1} x + C_2 \, \map {y_2} x$

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

where $\map W {y_1, y_2}$ denotes the Wronskian of $\map {y_1} x$ and $\map {y_2} x$.

Proof
Let the general solution of $(2)$ be:
 * $(3): \quad y = C_1 \, \map {y_1} x + C_2 \, \map {y_2} x$

Let the arbitrary constants $C_1$ and $C_2$ be replaced by functions $\map {v_1} x$ and $\map {v_2} x$.

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

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 $(2)$:
 * ${y_1} + \map P x {y_1}' + \map Q x y_1 = {y_2} + \map P x {y_2}' + \map Q x y_2 = 0$

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

In summary:

We started with the assumption that:
 * $(3): \quad y = C_1 \, \map {y_1} x + C_2 \, \map {y_2} x$

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:
 * $\map W {y_1, y_2} \ne 0$

and so $(11)$ is defined.

Thus:

and so as required:
 * $\displaystyle y = y_1 \int -\frac {\map {y_2} x \map R x} {\map W {y_1, y_2} } \rd x + y_2 \int \frac {\map {y_1} x \map R x} {\map W {y_1, y_2} } \rd x$

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 $\map {v_1} x$ and $\map {v_2} x$.