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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. Animportantfirststepis tonoticethatiff x andg x aretwo solutions,thensois thesum;infact,sois any linearcombinationA f x Bg x.

12.2 Behavior of the Solutions 179 Example 12.6 Find the solution y y x of y 2y 5y 0, with the initial values y 0 2 y 0 1. The auxiliary equation r2 2r 5 0 has the solutions r

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