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

Let us first rewrite the equation into

This equation will be called exact if

,

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

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

(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

The function 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 . Indeed, we have

,

which implies

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

(5)

Integrate to find ;

(6)

Write down the function F(x,y);

(7)

All the solutions are given by the implicit equation

(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.

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Last Update 6-24-98

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