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This is from Carmen Chicone's book on page 5, and I don't even know where to start. I haven't done differential equations in a while so this is really a problem for me. I'll accept any hints or suggestions.

Find the general solution of the differential equation $$\frac{dy}{dx} = \frac{y}{(y+2)e^y - 2x}.$$

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Hint: It is a linear equation $$\frac{1}{x'}=\frac{y}{(y+2)e^y - 2x} \\ x'=\frac{(y+2)e^y - 2x}{y}\\ x'+\frac{ 2}{y}x=(1+\frac{2 }{y})e^y$$ The factor is $$\mu=e^{\int\frac{2}{y}} =y^2$$ Then the solution is $$x=\frac{1}{y^2}[\int y^2(1+\frac{2 }{y})e^ydy+c]$$

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  • $\begingroup$ Yes the final answer was very satisfying! Thank you $\endgroup$ – Bob_Bobingston Mar 7 '14 at 22:15

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