Variation of Parameters

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Table of Contents

• Variation of Parameters
  • Key Definitions
  • Variation of Parameters
    • \(y''+py'+qy=0\)
    • \(W(t):=y_1y_2'-y_2y_1'\not = 0\)
    • \(y''+py'+qy=g(t)\)
    • \(y_p(t)=y_1\int\frac{-y_2(t)g(t)}{W(t)}dt+y_2\int\frac{y_1(t)g(t)}{W(t)}dt\)

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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 :

\(y''+py'+qy=0\)

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

\(W(t):=y_1y_2'-y_2y_1'\not = 0\)

Then the :

\(y''+py'+qy=g(t)\)

has a particular sol.:

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