TheAdjoint Equation. If a second order linear homogeneous

Chapter 3, Problem 46

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TheAdjoint Equation. If a second order linear homogeneous equation is not exact,it canbe made exact by multiplying by an appropriate integrating factor (x). Thus we requirethat (x) be such that(x)P(x)y+ (x)Q(x)y+ (x)R(x)y = 0can be written in the form[(x)P(x)y]+ [ f(x)y]= 0.By equating coefficients in these two equations and eliminating f(x), show that thefunction must satisfyP+ (2P Q)+ (P Q+ R) = 0.This equation is known as the adjoint of the original equation and is important inthe advanced theory of differential equations. In general, the problem of solving theadjoint differential equation is as difficult as that of solving the original equation, so onlyoccasionally is it possible to find an integrating factor for a second order equation.