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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.

Evaluating at x 0, we have 4 A B 1 A 5B. Solving this pair of equations, we get A 19 4 and B 3 4, so our solution is (12.13) y 19 4 e x 3 4 e 5x Example 12.4 A function x x t satisfies the differential equation (12.14) x 2x 15x 0 Under what conditions on the values of x at t 2 0 will this function decay to 0 as t ∞? The auxiliary equation r

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