<s>[INST] How do I solve the differential equation: $ \frac{dy}{dx} = \frac{x+y}{x-y} $? $$ \frac{dy}{dx} = \frac{x+y}{x-y}   $$
I have tried this problem so long... such as 
please help..
 [/INST] This is a dimensionally homogenous ODE. 
Let $u = \frac{y}{x}$
Then we have $\frac{du}{dx} = \frac{x\frac{dy}{dx} - y}{x^{2}}$
So $\frac{du}{dx} = \frac{\frac{dy}{dx} - u}{x}$
$\frac{dy}{dx} = x\frac{du}{dx} + u$
Substituting this in:
$x\frac{du}{dx} + u = \frac{x + ux}{x-ux}$
$x\frac{du}{dx} + u = \frac{1+u}{1-u}$
$x\frac{du}{dx} = \frac{1+u - u + u^{2}}{1-u}$
$x\frac{du}{dx} = \frac{u^{2} + 1}{1-u}$
$\int \frac{1-u}{u^{2}+1}du = \int \frac{1}{x} dx$
Can you solve from here?
</s>