An exact differential equation is a first order differential equation that has the form:
and is also exact. This means that:
To find the solution of this equation, we consider the fact that the equation has the form of an exact differential of a function φ(y,t). The total differential of φ(y,t) is then:
Comparing to our original differential equation, and and also that this implies that dφ(y,t)=0 and φ(y,t)=constant. To solve the equation, we integrate both sides of the total differential with respect to one of the variables, say t:
where F(y) is a "constant of integration". (In reality, F(y) is a function, but when we differentiate it with respect t, we treat y as constant and so its derivative becomes zero.) In order to solve for F(y), we differentiate φ(y,t) with respect to the other variable, in this case y, and set the result equal to N(t,y).
If the two functions, M(t,y) and N(t,y), instead have the property:
then the equation is said to be "inexact" and the above method cannot be used directly to solve it. However, it can be used if a function, μ(t,y) can be found such that, when the differential equation is multiplied by μ(t,y), it becomes exact. Such a function is called an "integrating function". This would transform the condition for exactness to:
meaning that the integrating function can be found as:
This isn't particularly useful by itself as we still have to solve for μ as a function of two variables. Often, the integrating function can be expressed as only one of the two variables, meaning we would only need to solve an ordinary differential equation rather than a partial differential equation. Otherwise, the integrating factor can sometimes be guessed or another method of finding a solution can be employed.
Consider the equation:
Here we have M(t,y)=3t+y and N(t,y)=t. As:
the equation is exact. Integrating M(t,y) with respect to t produces:
Therefore F'(y)=0 which means f(y) is a constant, a. Therefore the solution is:
where the constant a has been combined with the constant of the condition φ(t,y)=constant.
- K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 3rd ed., 2006
- Exact Equations from mathworld.wolfram.com