Given a simply connected and open subset D of R2 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
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 of a function. For a function , the exact or total derivative with respect to is given by
The function given by
is a potential function for the differential equation
Existence of potential functions
In physical applications the functions I and J are usually not only continuous but even continuously differentiable. Schwarz's Theorem then provides us with a necessary criterion for the existence of a potential function. For differential equations defined on simply connected sets the criterion is even sufficient and we get the following theorem:
Given a differential equation of the form (for example, when F has zero slope in the x and y direction at F(x,y)):
with I and J continuously differentiable on a simply connected and open subset D of R2 then a potential function F exists if and only if
Solutions to exact differential equations
Given an exact differential equation defined on some simply connected and open subset D of R2 with potential function F, a differentiable function f with (x, f(x)) in D is a solution if and only if there exists real number c so that
For an initial value problem
we can locally find a potential function by
for y, where c is a real number, we can then construct all solutions.
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
Combining the terms gives
If the equation is exact, then . Additionally, the total derivative of is equal to its implicit ordinary derivative . This leads to the rewritten equation
Now, let there be some second-order differential equation
If for exact differential equations, then
where is some arbitrary function only of that was differentiated away to zero upon taking the partial derivative of with respect to . Although the sign on could be positive, it is more intuitive to think of the integral's result as that is missing some original extra function that was partially differentiated to zero.
then the term should be a function only of and , since partial differentiation with respect to will hold constant and not produce any derivatives of . In the second order equation
only the term is a term purely of and . Let . If , then
Since the total derivative of with respect to is equivalent to the implicit ordinary derivative , then
Thus, the second order differential equation
is exact only if and only if the below expression
is a function solely of . Once is calculated with its arbitrary constant, it is added to to make . 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.
Now, however, in the final implicit solution there will be a term from integration of with respect to twice as well as a , two arbitrary constants as expected from a second-order equation.
Given the differential equation
one can always easily check for exactness by examining the term. In this case, both the partial and total derivative of with respect to are , so their sum is , which is exactly the term in front of . With one of the conditions for exactness met, one can calculate that
Letting , then
So, is indeed a function only of and the second order differential equation is exact. Therefore, and . Reduction to a first-order exact equation yields
Integrating with respect to yields
where is some arbitrary function of . Differentiating with respect to gives an equation correlating the derivative and the term.
So, and the full implicit solution becomes
Solving explicitly for 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
Implicit differentiation of the exact second-order equation times will yield an 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
is a function only of and . Combining all and terms not coming from gives
Thus, the three conditions for exactness for a third-order differential equation are: the term must be , the term must be and
must be a function solely of .
Consider the nonlinear third-order differential equation
If , then is and which together sum to . Fortunately, this appears in our equation. For the last condition of exactness,
which is indeed a function only of . So, the differential equation is exact. Integrating twice yields that . Rewriting the equation as a first-order exact differential equation yields
Integrating with respect to gives that . Differentiating with respect to and equating that to the term in front of in the first-order equation gives that
and that . The full implicit solution becomes
The explicit solution, then, is