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**Educated** $\displaystyle x' = x + y$

$\displaystyle y' = 2x$

Find an expression for $\displaystyle \frac{dy}{dx} $ in terms of x,y

Hint: Note $\displaystyle \frac{d}{dt}y(t) = \frac{d}{dt}y(x(t))=...$

Find an implicit solution of the differential equation in $\displaystyle x \mapsto y$ for y(x)

So, I don't know if I'm interpreting this right or not, but rewriting the equations in fraction notation:

$\displaystyle \frac{dx}{dt} = x + y$

$\displaystyle \frac{dy}{dt} = 2x$

I divide the second one by the first one to get:

$\displaystyle \frac{dy}{dt} \frac{dt}{dx} = \dfrac{2x}{x + y}$

$\displaystyle \frac{dy}{dx}= \dfrac{2x}{x + y}$

So I get an equation for dy/dx

And then I let y(x) = x v(x)

Which means y'(x) = v(x) + x v'(x)

$\displaystyle v(x) + x v'(x) = \dfrac{2x}{x + x v(x)}$

$\displaystyle v(x) + x v'(x) = \dfrac{2}{1 + v(x)}$

Have I done everything right so far?

Now what do I do here? I get this non-linear equation and I don't know how to solve it... any hints?