Variation of Parameters
For a second-order ordinary differential equation,
| y^('')+p(x)y^'+q(x)y=g(x). |
(1)
|
Assume that linearly independent solutions y_1(x) and y_2(x) are known to the homogeneous equation
| y^('')+p(x)y^'+q(x)y=0, |
(2)
|
and seek v_1(x) and v_2(x) such that
Now, impose the additional condition that
| v_1^'y_1+v_2^'y_2=0 |
(5)
|
so that
Plug y^*, y^*^', and y^*^('') back into the original equation to obtain
| v_1(y_1^('')+py_1^'+qy_1)+v_2(y_2^('')+py_2^'+qy_2)+v_1^'y_1^'+v_2^'y_2^'=g(x), |
(8)
|
which simplifies to
| v_1^'y_1^'+v_2^'y_2^'=g(x). |
(9)
|
Combing equations (◇) and (9) and simultaneously solving for v_1^' and v_2^' then gives
where
| W(y_1,y_2)=W(x)=y_1y_2^'-y_2y_1^' |
(12)
|
is the Wronskian, which is a function of x only, so these can be integrated directly to obtain
which can be plugged in to give the particular solution
| y^*=v_1y_1+v_2y_2. |
(15)
|
Generalizing to an nth degree ODE, let y_1, ..., y_n be the solutions to the homogeneous ODE and let v_1^'(x), ..., v_n^'(x) be chosen such that
and the particular solution is then
| y^*(x)=v_1(x)y_1(x)+...+v_n(x)y_n(x), |
(17)
|
See also
Ordinary Differential Equation, Second-Order Ordinary Differential EquationExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Variation of Parameters." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VariationofParameters.html