Exact differential equation

Exact differential equation
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 of 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 of a function. For a function , the exact or total derivative with respect to is given by
Example
The function given by
is a potential function for the differential equation
First order exact differential equations
Identifying first order exact differential equations
Let the functions , and , where the subscripts denote the partial derivative with respect to the relative variable, be continuous in the region . Then the differential equation
is exact if and only if
That is, there exists a function , called a potential function, such that
So, in general:
Proof
The proof has two parts.
First, suppose there is a function such that
It then follows that
Since and are continuous, then and are also continuous which guarantees their equality.
The second part of the proof involves the construction of and can also be used as a procedure for solving first order exact differential equations. Suppose that
and let there be a function for which
Begin by integrating the first equation with respect to . 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.
where is any differentiable function such that . The function plays the role of a constant of integration, but instead of just a constant, it is function of , since we is a function of both and and we are only integrating with respect to .
Now to show that it is always possible to find an such that .
Differentiate both sides with respect to .
Set the result equal to and solve for .
In order to determine from this equation, the right-hand side must depend only on . This can be proven by showing that its derivative with respect to is always zero, so differentiate the right-hand side with respect to .
Since ,
Now, this is zero based on our initial supposition that
Therefore,
And this completes the proof.
Solutions to first order exact differential equations
First order exact differential equations of the form
can be written in terms of the potential function
where
This is equivalent to taking the exact differential of .
The solutions to an exact differential equation are then given by
and the problem reduces to finding .
This can be done by integrating the two expressions and and then writing down each term in the resulting expressions only once and summing them up in order to get .
The reasoning behind this is the following. Since
it follows, by integrating both sides, that
Therefore,
where and are differentiable functions such that and .
In order for this to be true and for both sides to result in the exact same expression, namely , then must be contained within the expression for because it cannot be contained within , since it is entirely a function of and not and is therefore not allowed to have anything to do with . By analogy, must be contained within the expression .
Ergo,
for some expressions and .
Plugging in into the above equation, we find that
and so and turn out to be the same function. Therefore,
Since we already showed that
it follows that
So, we can construct by doing and and then taking the common terms we find within the two resulting expressions (that would be ) and then adding the terms which are uniquely found in either one of them - and .
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
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
and
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.
Next, if
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
So,
and
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.
Example
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
where
and where
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 .
Example
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
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.
Ordinary differential equations