To solve a non-exact differential equation, we need to first check if the given differential equation is exact or not. A differential equation of the form:

$M(x, y) \, dx + N(x, y) \, dy = 0$

is exact if the condition:

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

is satisfied. If this condition does not hold, the equation is non-exact.

In the case of a non-exact differential equation, we use an integrating factor (a function that makes the equation exact) to make it solvable.

Here are the general steps to solve a non-exact differential equation:

1. Check if the equation is exact by testing the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
2. Find an integrating factor: This could depend on $x$ or $y$, and there are methods to determine it (e.g., using a function of $x$ or $y$ or their combination).
3. Multiply the equation by the integrating factor to make it exact.
4. Solve the resulting exact equation using the standard method of solving exact differential equations.

If you provide a specific non-exact differential equation, I can walk you through solving it step by step. Could you share the equation you need help with?