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Second Order Linear Differential Equations

CHAPTER12 SecondOrderLinearDifferentialEquations a relationinvolvingvariablesx y y y . Asolutionis a functionf x suchthatthesubstitutiony f x y f x y f x givesanidentity. Thedifferentialequationissaidtobelineari f it is linearinthevariablesy y y . We have alreadyseen( )how tosolve firstorderlinearequations; ( )y ay by g x whereaandbareconstants,andg x is a differentiablefunctionofx. ,wesaw thata firstorderequationhasa one-parameterfamilyofsolutions,andthatth especificationofaninitialconditiony x0 y0uniquelydeterminesa , ,andnumbers y0 y 0, there isa uniquefunctionf x which solvesthedifferentialequation( )andsatisfiestheinitialconditionsf x0 y0 f x0 y tocompletelysolve equation( )whenthefunctionontherighthandsideis zero:( )y ay by 0 Thisis calledthehomogeneousequation.

Proposition 12.1 Let r be a root of the equation (12.9) r2 ar b 0 Then erx is a solution to the homogeneous equation: (12.10) y ay by 0 Equation (12.9) is called the auxiliary equation of the differential equation (12.10). To verify the propo-sition, let y erx so that y rerx y r2erx. Substituting into equation (12.10): (12.11) r2erx are rx berx ...

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