(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve [tex](x - \sqrt{xy})dy - ydx = 0[/tex]

Rearranged gives us

[tex]-y + (x - \sqrt{xy})y' = 0 [/tex]

And it looks like an exact differential equation, but is it really?

2. Relevant equations

For any given exact equation of the form

[tex]M(x,y) + N(x,y)y' = 0[/tex]

The following must be true

[tex]\frac{\partial}{\partial y}M(x,y) = \frac{\partial}{\partial x}N(x,y)[/tex]

Otherwise it is a nonexact equation, and then an integrating factor is needed, in order to make them exact.

3. The attempt at a solution

Let's check whether it is exact or nonexact,

[tex]\frac{\partial M}{\partial y} = -1[/tex]

[tex]\frac{\partial N}{\partial x} = 1 - \frac{y}{2\sqrt{xy}} [/tex]

So, as

[tex]\frac{\partial}{\partial y}M(x,y) \neq \frac{\partial}{\partial x}N(x,y)[/tex]

This happens to be a nonexact differential equation and therefore an integrating factor is needed, in order to turn it exact.

Now what I want to know, is how do find an integrating factor? Will it be a two variable function or a single variable function?

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# Homework Help: Solving Nonexact First Order ODEs.

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