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
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
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.
- Check that the equation is indeed exact;
- Write down the system
- 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.
- Use the second equation of the system to find the derivative
of . Indeed, we have
Note that is a function of
y only. Therefore, in the
the variable, x, should disappear.
Otherwise something went wrong!
- Integrate to find ;
- Write down the function F(x,y);
- All the solutions are given by the implicit equation
- If you are given an IVP, plug in the initial condition to find
the constant C.
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