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Question Number 115085 by Khalmohmmad last updated on 23/Sep/20

Commented by mohammad17 last updated on 23/Sep/20

$$x=vy\Rightarrow \frac{dx}{dy}=v+y\frac{dv}{dy}$$$$$$$$when:\frac{dx}{dy}=\frac{2e^{\frac{x}{y}}-2e^{\frac{x}{y}}\frac{x}{y}}{-1-2e^{\frac{x}{y}}}$$$$$$$$then:y\frac{dv}{dy}=\frac{2e^v-2{ve}^v+v+2{ve}^v}{-1-2e^v}$$$$$$$$\int \frac{dy}{y}=-\int \frac{2e^v+1}{2e^v+v}dv\Rightarrow lny=-ln\mid 2e^v+v\mid +lnc$$$$$$$$ln\mid 2{ye}^v+yv\mid =lnc\Rightarrow 2y{e}^v+yv=c$$$$$$$$put:v=\frac{x}{y}$$$$$$$$\therefore 2{ye}^{\frac{x}{y}}+x=c$$$$$$$$⟨moh⟩$$

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