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: {{begin-eqn}} {{eqn | l = y'       | r = \paren {v_1 {y_1}' + {v_1}' y_1} + \paren {v_2 {y_2}' + {v_2}' y_2}       | c = Product Rule for Derivatives }} {{eqn | n = 5       | r = \paren {v_1 {y_1}' + v_2 {y_2}'} + \paren { {v_1}' y_1 + {v_2}' y_2}       | c =  }} {{end-eqn}} 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: {{begin-eqn}} {{eqn | n = 7       | l = y'       | r = v_1 {y_1}' + v_2 {y_2}'       | c =  }} {{eqn | n = 8       | ll= \leadsto       | l = y''       | r = \paren {v_1 {y_1}'' + {v_1}' {y_1}'} + \paren {v_2 {y_2}'' + {v_2}' {y_2}'}       | c = Product Rule for Derivatives }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = y'' + \map P x y' + \map Q x y       | r = \map R x       | c = $(1):$ given }} {{eqn | ll= \leadsto       | l = \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       | r = \map R x       | c = substituting from $(8)$ }} {{eqn | ll= \leadsto       | l = \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       | r = \map R x       | c = substituting from $(7)$ }} {{eqn | ll= \leadsto       | l = \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}       | r = \map R x       | c = substituting from $(4)$ }} {{eqn | n = 9       | ll= \leadsto       | l = 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}'       | r = \map R x       | c = rearranging }} {{end-eqn}} 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: {{begin-eqn}} {{eqn | n = 6       | l = {v_1}' y_1 + {v_2}' y_2       | r = 0 }} {{eqn | n = 10       | l = {v_1}' {y_1}' + {v_2}' {y_2}'       | r = \map R x }} {{eqn | ll= \leadsto       | l = {v_1}'       | r = \frac {y_2 \map R x} {y_2 {y_1}' - y_1 {y_2}'}       | c =  }} {{eqn | l = {v_2}'       | r = \frac {y_1 \map R x} {y_1 {y_2}' - y_2 {y_1}'}       | c =  }} {{eqn | n = 11       | ll= \leadsto       | l = {v_1}'       | r = -\frac {y_2 \map R x} {\map W {y_1, y_2} }       | c =  }} {{eqn | l = {v_2}'       | r = \frac {y_1 \map R x} {\map W {y_1, y_2} }       | c =  }} {{end-eqn}} 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: {{begin-eqn}} {{eqn | l = v_1       | r = \int -\frac {y_2 \map R x} {\map W {y_1, y_2} } \rd x       | c =  }} {{eqn | l = v_2       | r = \int \frac {y_1 \map R x} {\map W {y_1, y_2} } \rd x       | c =  }} {{end-eqn}} 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 $\map {v_2} x$. == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x|next = Linear Second Order ODE/y'' + y = cosecant x}}: $\S 3.19$: The Method of Variation of Parameters