I did it a different way:
I haven't seen the notation you used before but I think i've written it down correctly. I also didn't use the integrating factor method. It just seemed easier to do it this way.
This is how you would do it with the integrating factor:
Thus, the integrating factor
Now, when you multiply the equation through by , the left side always becomes the derivative of the product between the dependent variable (in this case, ), and the integrating factor.
So we see that the equation now becomes
Integrate both sides and you get
Now multiply both sides by to get the solution
My solution and Showcase_22's are almost the same. However, he made a tiny mistake:
When you integrate, you get
His solution in the end would then be
Note that , another constant.
So we would then get the solution
They now match.
I hope this makes sense!