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 . 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.

two solutions, then so is the sum; in fact, so is any linear combination Af x Bg x . Thus, once we know two solutions (they must be independent in the sense that one isn’t a constant multiple of the other) we can solve the initial value problem in theorem 12.1 by solving for A and B. Example 12.1 Solve y y 0 y 0 4 y 0 1

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  Linear, Multiple, Second, Order, Differential, Equations, Second order linear differential equations

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