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 calledthehomog

12.1 Homogeneous Equations 177 Now unfortunately a quadratic equation does not necessarily always have two real roots, so we have to examine the cases separately. Case of two real roots. If the discriminant a2 4b 0, then there are two real roots, and it is straight-forward to find the solution of the corresponding initial value problem.

Loading..

Tags:

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

Information

Related search queries