Exact Equations
Before we begin, a quick note on notation. Within this section, subscripts of x and y mean "partial derivative with respect to x" and "partial derivative with respect to y" respectively. So
$ M_y(x,y) $
means the partial derivative of M(x,y) with respect to y.
Suppose, firstly, that your differential equation can be written this way:
$ M(x,y)+N(x,y)y' = 0 $
and secondly, that
$ \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} $
In this case*, your differential equation is said to be exact, and the solution to the differential equation is ψ(x,y) where
$ \psi_x(x,y)=M(x,y);\qquad \psi_y(x,y)=N(x,y) $
* There is another condition required for this technique to work that I did not want to mention initially for continuity reasons. The functions M, N, ∂M/∂y and ∂N/∂x must be continuous in a simply connected region R of the xy plane, and it must be true for every point in R that ∂M/∂y=∂N/∂x. By "simply connected", I simply mean that there are no holes in the region. Now, you should know this requirement, but most likely your professor will not try to trick you into using the technique mentioned on this page by giving you a problem in which R is not simply connected. So just take notice of this requirement and shelve it somewhere in your mind.
