Problem Description:

Solve the following variable coefficient 2^nd order equation by operator factorization:

Problem Solution:

1. First let’s write the original equation in operator form,

2. Now factor the operator L(D) as follows:

Checking this (just to be sure) shows that indeed this is the correct factorization (always be careful here), or

3. Now, let . Therefore, we have

or

But this is a linear 1^st order ODE with integrating factor . Therefore,

and integration gives

4. Now we must solve

or

Again this is a linear 1^st order ODE with integrating factor

Therefore,

Integrating both sides of this expression gives (see detailed integration by parts below):

or

Now let’s put this into the following form

where

Therefore, we have

or