Find a particular solution to in x> 0

so the characteristic solution is



As a guess for the particular solution, let's back up a bit. The reason the choice of works for the characteristic solution is that, in the background, we're employing the substitution , so that is getting replaced with a new function . Differentiating yields




Now the ODE in terms of is linear with constant coefficients, since the coefficients and will cancel, resulting in the ODE



Of coursesin, the characteristic equation will be , which leads to solutions , as before.

Now that we have two linearly independent solutions, we can easily find more via variation of parameters. If are the solutions to the characteristic equation of the ODE in terms of , then we can find another of the form where




where is the Wronskian of the two characteristic solutions. We have










and recalling that , we have