<s>[INST] How to separate variables in this equation: $\;y\frac{dy}{dx} = (x+7)(y^2+6)\;?$ I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$
I know that the first step is to isolate both term each side and then integrate...
But I can't figure out how to isolate term on this one, I would probably be able to solve the rest of the equation, just need to know how to start.
Thanks.
 [/INST] $$y\dfrac{dy}{dx}=(x+7)(y^2+6)\implies \dfrac{y}{y^2+6}\dfrac{dy}{dx}=(x+7)\implies \dfrac{y}{y^2+6} dy=(x+7) dx\implies \int\frac{y}{y^2+6}dy=\int x+7 \,dx\implies \frac12\log|y^2+6|=\dfrac{x^2}2+7x\implies \log|y^2+6|=x^2+14x+c\implies y^2+6=e^{x^2+14x+c}\implies y=\sqrt{e^{x^2+14x+c}-6}$$
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