8) use reduction of order to solve. one solution of homogeneo x2y" +7xy' +5y =x 1 x> 0 y1 = x here y1 is a solution of the corresponding homogeneous.

I suspect there's a typo in the question, because is *not* a solution to the corresponding homogeneous equation. We have and , so the ODE reduces to

Let , then and , and substituting these into the (homogeneous) ODE gives

which then admits the characteristic solutions and .

Now to find a solution to the non-homogeneous ODE. We look for a solution of the form or .

It doesn't matter which one we start with, so let's use the first case. We get derivatives and . Substituting into the ODE yields

Substitute , so that and

which is linear in , and we can condense the left side as the derivative of a product after multiplying both sides by :

Integrate to solve for :

Then multiply both sides by to solve for :

so we found another fundamental solution that satisifes this ODE.