This equation is not exact, since
So we look for an integrating factor
such that
For this to be exact, we require
Differentiating both sides gives
When we take
to be a function of either
or
, but not both, this partial differential equation reduces to one of the separable ordinary differential equations
both of which can be solved directly for
provided that the result on the right hand side of either ODE is a function of either only
or
, respectively.
The choice of which integrating factor
to look for is then decided by how easily the right hand side can be taken care of. We have
On the other hand, the integral resulting from an integrating factor
is more complicated/impossible to deal with. So, the integrating factor must be a function of
, which means
, and it satisfies
Distributing the integrating factor across the original ODE, we have
with partial derivatives
Thus the modified ODE is exact, as required. Now we try to find a solution of the form
.
Therefore the general solution is