# Thread: Interesting Equation

1. ## Interesting Equation

(xe^(y/x)+ 2(x^2)y)dy − (ye^(y/x))dx = 0

I've tried solving this equation with no luck. I have tried a variety of different ways but I end up with integrals that are impossible to solve.

Any suggestions would be greatly appreciated. Thanks!

2. Originally Posted by FatherMike
(xe^(y/x)+ 2(x^2)y)dy − (ye^(y/x))dx = 0

I've tried solving this equation with no luck. I have tried a variety of different ways but I end up with integrals that are impossible to solve.

Any suggestions would be greatly appreciated. Thanks!
If you group the terms as follows

$\displaystyle (x \,dy - y \,dx) e^{y/x} + 2y\,dy = 0$

then divide by $\displaystyle x^2$ so

$\displaystyle \frac{x\,dy - y\,dx}{x^2} e^{y/x} + 2y \, dy = 0$

so

$\displaystyle d\left(\frac{y}{x} \right) e^{y/x} + d\left(y^2\right) = 0$

which can be integrated giving

$\displaystyle e^{y/x} + y^2 = c.$