# Separable form by substitution

Please suggest appropriate substitution to reduce it to separable form

$$\frac{dy}{dx} = \frac{4x+7y+2}{4x+7y+3}$$

let $$z=4x+7y$$ then $$\frac{dz}{dx}=4+7\frac{dy}{dx}$$ $$\frac {dy}{dx}= \frac17 (\frac{dz}{dx}-4)$$ $$\frac 17 (\frac {dz}{dx}-4) = \frac {z+2}{z+3}$$ please help further now i need the value of $$\frac {dz}{dx}$$

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## 1 Answer

Let $u=4x+7y$, now we get: $du/dx=4+7dy/dx$

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When you have $$\frac{dy}{dx}=f\bigg(\frac{ax+by+c}{a'x+b'y+c'}\bigg)$$ and $\frac{a}{a'}=\frac{b}{b'}$, so you can take $ax+by+c=t$ as a substitution. After that find $a'x+b'y+c'$ according to $t$ and solve the resulted ODE. – Babak S. Nov 15 '12 at 15:05
It is customary to take whole $4x+7y+2$ as $u$. – Babak S. Nov 15 '12 at 15:09
Thanks @BabakSorouh for detailed explanation :) – TPSstar Nov 15 '12 at 15:19
Furthermore i stuck here $$\frac 17 (\frac {dz}{dx}-4) = \frac {z+2}{z+3}$$ I need the value of $$\frac {dz}{dx}$$ – TPSstar Nov 15 '12 at 15:34
@TPSstar: See my second comment above and do the same with that substitution again. – Babak S. Nov 15 '12 at 17:17