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. 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 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 The function given by is a potential function for the differential equation 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: 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 .