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Second Order Linear Differential Equations - Math

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CHAPTER12SecondOrderLinearDifferentialEq uations 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 . Thus,onceweknow two solutions(they mustbeindependentin thesensethatoneisn t a constantmultipleoftheother)wecansolve y 0 y 0 4 y 0 1 Now, weknow thatcosxandsinxaresolutionsoftheequation ,sowetrya solutionoftheformy x Acosx Bsinx.

Second Order Linear Differential Equations 12.1. Homogeneous Equations A differential equation is a relation involvingvariables x y y y . A solution is a function f x such that the substitution y f x y f x y f x gives an identity. The differential equation is

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