# Thread:

1. ## Is this differential equation separable?

Is this separable? I'm trying, but I'm not sure if I'm on the right track:

y' = (x+y)/(x+2y)

dy/dx = (x+y)/(x+2y)

multiply both sides by dx:

dy = (x+y)/(x+2y) dx

Now I am not sure what to do because if I get (x+2y) to the other side, it'd still have an x and would be attached to the dy as well.

2. ## Re: Is this differential equation separable?

Is this separable? I'm trying, but I'm not sure if I'm on the right track:

y' = (x+y)/(x+2y)

dy/dx = (x+y)/(x+2y)

multiply both sides by dx:

dy = (x+y)/(x+2y) dx

Now I am not sure what to do because if I get (x+2y) to the other side, it'd still have an x and would be attached to the dy as well.
I'd do it like this... Make the substitution $\displaystyle \begin{align*} v = \frac{y}{x} \implies y = v\,x \implies \frac{dy}{dx} = v + x\,\frac{dv}{dx} \end{align*}$, then

$\displaystyle \begin{align*} \frac{dy}{dx} &= \frac{x + y}{x + 2y} \\ \frac{dy}{dx} &= \frac{1 + \frac{y}{x}}{1 + 2\left(\frac{y}{x}\right)} \\ v + x\,\frac{dv}{dx} &= \frac{1 + v}{1 + 2v} \\ x\,\frac{dv}{dx} &= \frac{1 + v}{1 + 2v} - v \\ x\,\frac{dv}{dx} &= \frac{1 - 2v^2}{1 + 2v} \\ \frac{1 + 2v}{1 - 2v^2}\,\frac{dv}{dx} &= \frac{1}{x} \end{align*}$

Go from here.
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