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I hope this helps you
Let's check if the ODE is exact. To do that, we want to show that if
then
. We have
so the equation is indeed exact. We're looking for a solution of the form
. Computing the total differential yields the original ODE,
Integrate both sides of the first PDE with respect to
; then
where
is a function of 
alone. Differentiate this with respect to so that
So the solution to this ODE is
i.e.
then
so the equation is indeed exact. We're looking for a solution of the form
Integrate both sides of the first PDE with respect to
where
So the solution to this ODE is
i.e.