I've got a differential equation:
e^x*y*y'=e^(-y)+e^(-2x-y)

I found an implicit solution:
y^2(e^y)=-4e^(-x)-(4/3)e^(-3x)+c

Could you guys check this? If this is right, is there a way to get an explicit solution?

Question

I've got a differential equation:
e^x*y*y'=e^(-y)+e^(-2x-y)

I found an implicit solution:
y^2(e^y)=-4e^(-x)-(4/3)e^(-3x)+c

Could you guys check this? If this is right, is there a way to get an explicit solution?

anonymous · Answer

e^(x)(y)(y')

anonymous · Answer

I would say:
$$e^xyy'=e^{-y}+e^{-2x}e^{-y} \implies e^xy \frac{dy}{dx}=e^{-y}(1+e^{-2x})$$
$$e^yydy=e^{-x}(1+e^{-2x})dx$$
Integrating:
$$\int\limits y e^y dy=\int\limits (e^{-x}+e^{-3x})dx \implies ye^y-\int\limits e^ydy=-e^{-x}-\frac{1}{3}e^{-3x}+C \implies ye^y-e^y=$$$$-e^{-x}-\frac{1}{3}e^{-3x}+C$$

anonymous · Answer

What did you do?

anonymous · Answer

I got (e^y)(y^2)/4 on the left side, and multiplied through by 4.

anonymous · Answer

Ok, I did the wrong integration by parts.

anonymous · Answer

I did mine in my head so mine may be wrong D: