Problem 114
Find the general integral of the total differential equation…
Find the general integral of the total differential equation $xy'\cos y+\sin y=0$.
Explanation of Method
Write in the form $M\,dx+N\,dy=0$ with $M=\sin y$, $N=x\cos y$; check exactness and integrate.
Step 1: Check exactness
$M_y=\cos y=N_x$, so the equation is exact.
Step 2: Find the potential function
$F=\int\sin y\,dx=x\sin y+g(y)$. Matching $F_y=x\cos y+g'(y)=N=x\cos y$ gives $g'(y)=0$.
Result
Final answer:
\[\boxed{\,x\sin y=C,\qquad C\in\mathbb{R}}\]