Find the general solution to the ODE
The ODE is clearly not separable.
Therefore the ODE is not exact.
Assume that an integrating factor exists as a function of x only.
The exact ODE is therefore
We seek a potential function u(x, y) such that
It does not take long to deduce that the potential function must be
Therefore the general solution of the ODE is
or, equivalently,
Find the general solution to the ODE
The ODE is clearly not separable, but it is linear.
The general solution is therefore
It is easy to check that this solution is correct:
Also note that the original ODE happens to be “exact” (in the sense that the complete left hand side can be rewritten as a single derivative):
from which the general solution follows immediately.
