Find all the solutions to
Solution: Note that this equation is in fact homogeneous. But let us use the technique of exact and nonexact to solve it. Let us follow these steps:
Hence, and .
,
which clearly implies that the equation is not exact.
.
Therefore, an integrating factor u(x) exists and is given by
,
which is exact. (Check it!)
,
which implies , that is, is constant. Therefore, the function F(x,y) is given by
We don't have to keep the constant C due to the nature of the solutions (see next step).
Remark: Note that if you consider the function
,
then we get another integrating factor for the same equation. That is, the new equation
is exact. So, from this example, we see that we may not have
uniqueness of the integrating factor. Also, you may learn that if the
integrating factor is given to you, the only thing you have to do is
multiply your equation and check that the new one is exact.
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Author: Mohamed Amine Khamsi