Variation of Parameters

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Table of Contents

• Variation of Parameters
  • Key Definitions
  • Variation of Parameters
    • $y^{″}+p{y}^{\prime }+qy=0$
    • $W(t):=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }\ne 0$
    • $y^{″}+p{y}^{\prime }+qy=g(t)$
    • $y_p(t)=y_1\int \frac{-y_2(t)g(t)}{W(t)}dt+y_2\int \frac{y_1(t)g(t)}{W(t)}dt$

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Variation of Parameters

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Key Definitions

Variation of Parameters - a method to find a particular solution to when is not possible i.e. $p,q$ are not constant or $g(t)$ is not “simple”

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Variation of Parameters

Given the fundamental set of solutions $y_1,y_2$ to a :

$y^{″}+p{y}^{\prime }+qy=0$

s.t. $y_1,y_2$ are lin. indep. as given by the :

$W(t):=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }\ne 0$

Then the :

$y^{″}+p{y}^{\prime }+qy=g(t)$

has a particular sol.:

$y_p(t)=y_1\int \frac{-y_2(t)g(t)}{W(t)}dt+y_2\int \frac{y_1(t)g(t)}{W(t)}dt$