Exact differential equation
An exact differential equation ( also complete) is an ordinary differential equation of the form
Wherein there is a continuously differentiable function, so is considered that
Such a function is then called the potential function of the vector field.
Existence of a potential function
Are and continuously partial differentiable and is the domain of and a simply connected domain in, then there exists then such a potential function, when the so-called integrability condition
The two-dimensional vector field can then be represented as the gradient of the potential:
The above integrability condition means that if the rotation of the vector field vanishes on a simply connected domain, a potential exists. In addition, twice continuously differentiable ( set of black ):
If the vector field has a potential function, then the differential equation is the total derivative of by:
Thus there must be to each solution of the above differential equation is a constant, so that the implicit equation
Is satisfied. Sometimes you can resolve this implicit equation explicitly. In this respect, this implicit version is a first step to explicitly solving the differential equation. For this reason is referred to as a first integral of the differential equation.
For an ordinary differential equation of the form, which does not meet the requirement, can be occasionally a null set free continuously differentiable function determined such that
An exact differential equation. In this case is referred to as an integral factor or as Euler's multiplier. Since by definition never zero, this differential equation, the same solutions as before multiplication has.
It is precisely then an integrating factor if the partial differential equation
It is usually difficult, this partial differential equation to solve in general. But since you only need a special solution, you will try to find with specific approaches to a solution. Such approaches could be for example:
Such an approach usually leads then to succeed if they are the partial differential equation turns into an ordinary differential equation.
There is an integral factor, so there is a potential applies:
Criterion for the existence of integrating factors
A general criterion for the existence of an integrating factor, in the case of an exact differential equation provides the Frobeniussche integrability.
Example of a specific criterion
There are many criteria for the existence of an integral molding of certain factors. Prototypical of this is for example the following for an integrating factor:
The domain of the vector field is a simply connected domain of. If there is a function so that
Holds, then every non-trivial solution is
An integrating factor.
For the integrability
Equivalent to ( product rule )
In other words
Since the zero function is a solution of, have any other solutions according to the uniqueness theorem no zeros. Thus, an integrating factor.