Problem 66
Find the general integral of the equation dx+(e^y-x) dy=0
Find the general integral of the equation $dx+(e^y-x)\,dy=0$.
Explanation of Method
Treat $x$ as the unknown function of $y$: $\dfrac{dx}{dy}-x=-e^y$ is linear in $x$, solved with the integrating factor $\mu(y)=e^{-y}$.
Step 1: Rewrite with $x=x(y)$
Dividing by $dy$: $\dfrac{dx}{dy}+(e^y-x)=0$, i.e. $\dfrac{dx}{dy}-x=-e^y$.
Step 2: Apply the integrating factor
With $\mu(y)=e^{-y}$: $\dfrac{d}{dy}(e^{-y}x)=-1$, so $e^{-y}x=-y+C$.
Result
Final answer:
\[\boxed{\,x=Ce^{y}-ye^{y},\qquad C\in\mathbb{R}}\]