In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.

Definition

Given a simply connected and open subset D and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form

is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that

and

An exact equation may also be presented in the following form:

where the same constraints on I and J apply for the differential equation to be exact.

The nomenclature of "exact differential equation" refers to the exact differential, the exact or total derivative is given by

Example

given by

is a potential function for the differential equation

First-order exact differential equations

Identifying first-order exact differential equations

, where the subscripts denote the partial derivative. Then the differential equation

is exact if and only if

, called a potential function, such that

So, in general:

Proof

The proof has two parts.

are also continuous which guarantees their equality.

. In practice, it doesn't matter if you integrate the first or the second equation, so long as the integration is done with respect to the appropriate variable.

.

And this completes the proof.

Solutions to first-order exact differential equations

This is equivalent to taking the total derivative

.

.

.

must must.

.

Second-order exact differential equations

The concept of exact differential equations can be extended to second-order equations. Consider starting with the first-order exact equation:

Since both functions are functions of two variables, implicitly differentiating the multivariate function yields

Expanding the total derivatives gives that

and that

terms gives

If the equation is exact, then . This leads to the rewritten equation

Now, let there be some second-order differential equation

for exact differential equations, then

and

that was partially differentiated to zero.

Next, if

. In the second-order equation

, then

, then

So,

and

Thus, the second-order differential equation

and only if the below expression

. If the equation is exact, then we can reduce to the first-order exact form which is solvable by the usual method for first-order exact equations.

, two arbitrary constants as expected from a second-order equation.

Example

Given the differential equation

. With one of the conditions for exactness met, one can calculate that

, then

. Reduction to a first-order exact equation yields

yields

term.

and the full implicit solution becomes

yields

Higher-order exact differential equations

The concepts of exact differential equations can be extended to any order. Starting with the exact second-order equation

it was previously shown that equation is defined such that

th-order differential equation with new conditions for exactness that can be readily deduced from the form of the equation produced. For example, differentiating the above second-order differential equation once to yield a third-order exact equation gives the following form

where

gives

and

.

Example

Consider the nonlinear third-order differential equation

. Fortunately, this appears in our equation. For the last condition of exactness,

. Rewriting the equation as a first-order exact differential equation yields

. The full implicit solution becomes

The explicit solution, then, is

See also

• Exact differential

• Inexact differential equation

References

Further reading

• Boyce, William E.; DiPrima, Richard C. (1986). Elementary Differential Equations (4th ed.). New York: John Wiley & Sons, Inc.