8.13. Special Methods For Determining A Particular Solution Of The Nonhomogeneous Equation.  Reduction To A System Of First-Order Linear Equations

Although the method variation of parameters can always provide a particular solution to the nonhomogeneous equation, it may not be the simplest way to do so in specific cases.

Example

Find a particular solution of

        ( D1 )( D2 )y=x e x+ x 2                 (8.25)

Solution

Let u=( D2 )y .  Then (8.25) becomes

          ( D1 )u=x e x+ x 2

which is a 1st order equation that can be solved using Theorem 8.1 or by inspection.  This gives

        u= 1 2 e x+ x 2

so that

        ( D2 )y=u = 1 2 e x+ x 2

Using Theorem 8.1, we have, with y 1 ( 0 )=0 ,

        y 1 ( x )= 1 2 e 2x 0 x dt    e t 2 t

Even though the integral cannot be evaluated in terms of elementary functions, we can consider the problem solved.  The general solution is

        y( x )= c 1 e x + c 2 e 2x + 1 2 e 2x 0 x dt    e t 2 t