Exact differential.

• Sep 11th 2010, 11:10 PM
ceode
Exact differential.
(1+e^x/y)dx + e^x/y ( 1-x/y )dy =0
• Sep 11th 2010, 11:14 PM
Prove It
Is this

$\displaystyle \left(1 + \frac{e^x}{y}\right)\,dx + \frac{e^x}{y}\left(1 - \frac{x}{y}\right)\,dy = 0$

or

$\displaystyle \left(1 + e^{\frac{x}{y}}\right)\,dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)\,dy = 0$?
• Sep 11th 2010, 11:20 PM
ceode
2nd one
• Sep 12th 2010, 01:00 AM
Prove It
$\displaystyle \displaystyle{\left(1 + e^{\frac{x}{y}}\right)\,dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)\,dy = 0}$

If this is an exact equation of the form $\displaystyle M\,dx + N\,dy = 0$ then $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

$\displaystyle \frac{\partial M}{\partial y} = -\frac{x}{y^2}e^{\frac{x}{y}}$.

$\displaystyle \frac{\partial N}{\partial x} = -\frac{1}{y}e^{\frac{x}{y}} + \frac{1}{y}e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)$

$\displaystyle = -\frac{1}{y}e^{\frac{x}{y}} + \frac{1}{y}e^{\frac{x}{y}} - \frac{x}{y^2}e^{\frac{x}{y}}$

$\displaystyle = -\frac{x}{y^2}e^{\frac{x}{y}}$

$\displaystyle = \frac{\partial M}{\partial y}$ as required.

So it's an exact equation.

That means that the solution $\displaystyle F(x, y)$ has the property that

$\displaystyle \frac{\partial F}{\partial x} = M$ and $\displaystyle \frac{\partial F}{\partial y} = N$.

Thus $\displaystyle \frac{\partial F}{\partial x} = 1 + e^{\frac{x}{y}}$

$\displaystyle F = \int{1 + e^{\frac{x}{y}}\,dx}$

$\displaystyle = x + y\,e^{\frac{x}{y}} + g(y)$.

$\displaystyle \frac{\partial F}{\partial y} = e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)$

$\displaystyle F = \int{e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)\,dy}$

$\displaystyle = y\,e^{\frac{x}{y}} + h(x)$ (according to Wolfram)...

Comparing these two results, it's easy to see that $\displaystyle h(x) = x + C$ and $\displaystyle g(y) = C$.

Therefore $\displaystyle F =x + y\,e^{\frac{x}{y}} + C$.
• Sep 15th 2010, 03:41 AM
ceode
Thank you very much