Method of Variation of Parameters

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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:

$\ds y = y_1 \int -\frac {\map {y_2} x \map R x} {\map W {y_1, y_2} } \rd x + y_2 \int \frac {\map {y_1} x \map R x} {\map W {y_1, y_2} } \rd x$

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:

\(\ds y'\) \(=\) \(\ds \paren {v_1 {y_1}' + {v_1}' y_1} + \paren {v_2 {y_2}' + {v_2}' y_2}\) Product Rule for Derivatives
\(\text {(5)}: \quad\) \(\ds \) \(=\) \(\ds \paren {v_1 {y_1}' + v_2 {y_2}'} + \paren { {v_1}' y_1 + {v_2}' y_2}\)


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:

\(\text {(7)}: \quad\) \(\ds y'\) \(=\) \(\ds v_1 {y_1}' + v_2 {y_2}'\)
\(\text {(8)}: \quad\) \(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds \paren {v_1 {y_1} + {v_1}' {y_1}'} + \paren {v_2 {y_2} + {v_2}' {y_2}'}\) Product Rule for Derivatives


Hence:

\(\ds y + \map P x y' + \map Q x y\) \(=\) \(\ds \map R x\) $(1):$ given
\(\ds \leadsto \ \ \) \(\ds \paren {v_1 {y_1} + {v_1}' {y_1}'} + \paren {v_2 {y_2} + {v_2}' {y_2}'} + \map P x y' + \map Q x y\) \(=\) \(\ds \map R x\) substituting from $(8)$
\(\ds \leadsto \ \ \) \(\ds \paren {v_1 {y_1} + {v_1}' {y_1}'} + \paren {v_2 {y_2} + {v_2}' {y_2}'} + \map P x \paren {v_1 {y_1}' + v_2 {y_2}'} + \map Q x y\) \(=\) \(\ds \map R x\) substituting from $(7)$
\(\ds \leadsto \ \ \) \(\ds \paren {v_1 {y_1} + {v_1}' {y_1}'} + \paren {v_2 {y_2} + {v_2}' {y_2}'} + \map P x \paren {v_1 {y_1}' + v_2 {y_2}'} + \map Q x \paren {v_1 y_1 + v_2 y_2}\) \(=\) \(\ds \map R x\) substituting from $(4)$
\(\text {(9)}: \quad\) \(\ds \leadsto \ \ \) \(\ds v_1 \paren { {y_1} + \map P x {y_1}' + \map Q x y_1} + v_2 \paren { {y_2} + \map P x {y_2}' + \map Q x y_2} + {v_1}' {y_1}' + {v_2}' {y_2}'\) \(=\) \(\ds \map R x\) rearranging


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:

\(\text {(6)}: \quad\) \(\ds {v_1}' y_1 + {v_2}' y_2\) \(=\) \(\ds 0\)
\(\text {(10)}: \quad\) \(\ds {v_1}' {y_1}' + {v_2}' {y_2}'\) \(=\) \(\ds \map R x\)
\(\ds \leadsto \ \ \) \(\ds {v_1}'\) \(=\) \(\ds \frac {y_2 \map R x} {y_2 {y_1}' - y_1 {y_2}'}\)
\(\ds {v_2}'\) \(=\) \(\ds \frac {y_1 \map R x} {y_1 {y_2}' - y_2 {y_1}'}\)
\(\text {(11)}: \quad\) \(\ds \leadsto \ \ \) \(\ds {v_1}'\) \(=\) \(\ds -\frac {y_2 \map R x} {\map W {y_1, y_2} }\)
\(\ds {v_2}'\) \(=\) \(\ds \frac {y_1 \map R x} {\map W {y_1, y_2} }\)


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:

\(\ds v_1\) \(=\) \(\ds \int -\frac {y_2 \map R x} {\map W {y_1, y_2} } \rd x\)
\(\ds v_2\) \(=\) \(\ds \int \frac {y_1 \map R x} {\map W {y_1, y_2} } \rd x\)


and so as required:

$\ds y = y_1 \int -\frac {\map {y_2} x \map R x} {\map W {y_1, y_2} } \rd x + y_2 \int \frac {\map {y_1} x \map R x} {\map W {y_1, y_2} } \rd x$

$\blacksquare$


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 $\map {v_2} x$.


Sources

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