Let
be a region in
and let the functions\,
,\,
have continuous partial derivatives in
.\, The first order differential equation
or
![{\displaystyle {\begin{matrix}X(x,\,y)dx+Y(x,\,y)dy=0\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55bc9fba7151d94bb26f7dae159ad2f963fb53f6)
is called an exact differential equation , if the condition
is true in
.
Then there is a function\,
\, such that the equation (1) has the form
whence its general integral is
The solution function
can be calculated as the line integral
![{\displaystyle {\begin{matrix}f(x,\,y):=\int _{P_{0}}^{P}[X(x,\,y)\,dx+Y(x,\,y)\,dy]\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec362196c4c8579deb0e51444ee681d4d702c253)
along any curve
connecting an arbitrarily chosen point \,
\, and the point\,
\, in the region
(the integrating factor is now
).\\
Example. \, Solve the differential equation
This equation is exact, since
If we use as the integrating way the broken line from\,
\, to\,
\, and from this to\,
,\, the integral (2) is simply
Thus we have the general integral
of the given differential equation.