Exact First-Order Ordinary Differential Equation

Consider a first-order ODE in the slightly different form

Such an equation is said to be exact if

This statement is equivalent to the requirement that a conservative field exists, so that a scalar potential can be defined. For an exact equation, the solution is

where is a constant.

A first-order ODE (◇) is said to be inexact if

For a nonexact equation, the solution may be obtained by defining an integrating factor of (◇) so that the new equation

satisfies

or, written out explicitly,

This transforms the nonexact equation into an exact one. Solving (7) for gives

Therefore, if a function satisfying (8) can be found, then writing

in equation (◇) then gives

which is then an exact ODE. Special cases in which can be found include -dependent, -dependent, and -dependent integrating factors.

Given an inexact first-order ODE, we can also look for an integrating factor so that

For the equation to be exact in and , the equation for a first-order nonexact ODE

becomes

Solving for gives

which will be integrable if

in which case

so that the equation is integrable

and the equation

with known is now exact and can be solved as an exact ODE.

Given an exact first-order ODE, look for an integrating factor . Then

Combining these two,

For the equation to be exact in and , the equation for a first-order nonexact ODE

becomes

Therefore,

Define a new variable

then , so

Now, if

then

so that

and the equation

is now exact and can be solved as an exact ODE.

Given an inexact first-order ODE, assume there exists an integrating factor

so . For the equation to be exact in and , equation (◇) becomes

Now, if

then

so that

and the equation

is now exact and can be solved as an exact ODE.

Given a first-order ODE of the form

define

Then the solution is

If

where

then letting

gives

This can be integrated by quadratures, so