Definition

A differential equation of is said to be exact if it can be written in the form ${\displaystyle M(x,y)dx+N(x,y)dy=0}$ where ${\displaystyle M}$ and ${\displaystyle N}$ have continuous partial derivatives such that ${\displaystyle {\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}}$.

Solution

Solving the differential equation consists of the following steps:

1. Create a function ${\displaystyle f(x,y):=\int M(x,y)dx}$. While integrating, add a constant function ${\displaystyle g(y)}$ that is a function of ${\displaystyle y}$. This is a term that becomes zero if function ${\displaystyle f(x,y)}$ is differentiated with respect to ${\displaystyle x}$.
2. Differentiate the function ${\displaystyle f(x,y)}$ with respect to ${\displaystyle {\frac {\partial f}{\partial y}}}$. Set ${\displaystyle {\frac {\partial f}{\partial y}}=N(x,y)}$. Solve for the function ${\displaystyle g(y)}$.