Exact and Nonexact Equations

All the techniques we have reviewed so far were not of a general nature since in each case the equations themselves were of a special form. So, we may ask, what to do for the general equation

displaymath46

Let us first rewrite the equation into

displaymath40

This equation will be called exact if

displaymath41,

and nonexact otherwise. The condition of exactness insures the existence of a function F(x,y) such that

displaymath50

When the equation (E) is exact, we solve it using the following steps:

(1)
Check that the equation is indeed exact;
(2)
Write down the system

displaymath50

(3)
Integrate either the first equation with respect of the variable x or the second with respect of the variable y. The choice of the equation to be integrated will depend on how easy the calculations are. Let us assume that the first equation was chosen, then we get

displaymath42

The function tex2html_wrap_inline58 should be there, since in our integration, we assumed that the variable y is constant.

(4)
Use the second equation of the system to find the derivative of tex2html_wrap_inline58 . Indeed, we have

displaymath64,

which implies

displaymath66

Note that tex2html_wrap_inline68 is a function of y only. Therefore, in the expression giving tex2html_wrap_inline58 the variable, x, should disappear. Otherwise something went wrong!

(5)
Integrate to find tex2html_wrap_inline58;
(6)
Write down the function F(x,y);
(7)
All the solutions are given by the implicit equation

displaymath80

(8)
If you are given an IVP, plug in the initial condition to find the constant C.

You may ask, what do we do if the equation is not exact? In this case, one can try to find an integrating factor which makes the given differential equation exact.

[Differential Equations] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi
Last Update 6-24-98

Copyright 1999-2014 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour