How to Solve EXACT Differential Equations

Definition of Exact Equation

A differential equation of type

Mdx+Ndy=0 where M and N are all functions of x

is called an exact differential equation if there exists a function of two variables F(x,y) with continuous partial derivatives such that

dF(x,y) = Mdx+Ndy.

The general solution of an exact equation is given by

F(x,y)=C,

where C is an arbitrary constant.

Test for Exactness

Let functions M and N have continuous partial derivatives in a certain domain D. The differential equation Mdx+Ndy =0 is an exact equation if and only if

∂M/∂y=∂N/∂x.

Algorithm for Solving an Exact Differential Equation

First it’s necessary to make sure that the differential equation is exact using the test for exactness:

∂M/∂y=∂N/∂x.

Then we write the system of two differential equations that define the function F(x,y):

∂F/∂x=M and ∂F/∂y=N

Integrate the first equation over the variable x. Instead of the constant C, we write an unknown function of y:

F(x,y) = ∫M(x,y)dx.

The we differentiating with respect to y, we substitute the function F(x,y)into the second equation:

∂F/∂y = ∂/∂y[∫M(x,y)dx+φ(y)]=N(x,y).

From here we get expression for the derivative of the unknown function φ(y):

φ′(y)=M−∂F/∂y

By integrating the last expression, we find the function φ(y) and, hence, the function F(x,y):

F(x,y) = ∫Mdx+φ(y).

The general solution of the exact differential equation is given by

F(x,y)=C.