Let y=y(x) solve (x+2)e^(y+1)/(x+2),dx + (y+1),dx = (x+2),dy, y(1)=1.

Question

Let $y=y(x)$ solve $(x+2)e^{\frac{y+1}{x+2}}\,dx + (y+1)\,dx = (x+2)\,dy$, $y(1)=1$. If the domain of $y=y(x)$ is $(\alpha,\beta)$, then $|\alpha+\beta|$ is \_\_\_\_\_\_.

Answer: 4

Accepted Answer

Substituting $Y=y+1,X=x+2$ gives $-e^{-Y/X}=\ln|X|+c$. Using $(3,2)	o(1,1)$ translated: solving gives the domain endpoints summing (in absolute value) to 4.

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