Problem 120
Find the general integral of the total differential equation…
Find the general integral of the total differential equation $e^y\,dx+(xe^y-2y)\,dy=0$.
Explanation of Method
Check exactness with $M=e^y$, $N=xe^y-2y$, and integrate.
Step 1: Check exactness
$M_y=e^y=N_x$, so the equation is exact.
Step 2: Find the potential function
$F=\int e^y\,dx=xe^y+g(y)$. Matching $F_y=xe^y+g'(y)=xe^y-2y$ gives $g'(y)=-2y$, so $g(y)=-y^2$.
Result
Final answer:
\[\boxed{\,xe^y-y^2=C,\qquad C\in\mathbb{R}}\]