The roots of the characteristic equation are the solutions to this problem.
If the roots don't present in nice form, like in the case of this equation, you can put them into the quadratic equation.
Just taking the posative root for now
Well....I'm not sure how to simplify this but this is one of our solutions (e to the power of this).
Reduction of order might be best in this case. But to be honest, I'm not a fan of the way your notation it's kinda hard to read. Also, reduction of order problems typically already give you one solution so that you can find the other. Are you asked to find in general the reduction of order for this quesstion (i.e. both y1 and y2 without being given either of them)?
I also don't recognize how you're doing this. You multiply by Y1 and Y2 and then subtract the equations? Generally, reduction of order is
Then we differentiate here and sub back into our equation. We can then find a linear representation for this equation and find a solution.