Exact differential equation

Summary

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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 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,[1][2] 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

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

Solving

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.[3] 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

References

  1. ^ Wolfgang Walter (11 March 2013). Ordinary Differential Equations. Springer Science & Business Media. ISBN 978-1-4612-0601-9.
  2. ^ Vladimir A. Dobrushkin (16 December 2014). Applied Differential Equations: The Primary Course. CRC Press. ISBN 978-1-4987-2835-5.
  3. ^ Tenenbaum, Morris; Pollard, Harry (1963). "Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method.". Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering and the Sciences. New York: Dover. pp. 248. ISBN 0-486-64940-7.

Further reading

  • Boyce, William E.; DiPrima, Richard C. (1986). Elementary Differential Equations (4th ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-07894-8