# Differential equations/Exact differential equations

 Educational level: this is a tertiary (university) resource.
 Type classification: this is a lesson resource.
 Subject classification: this is a mathematics resource.
 Completion status: this resource is ~25% complete.

### 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 ${\displaystyle g(y)}$ .