Problem 121
Find the general integral of the total differential equation…
Find the general integral of the total differential equation $yx^{y-1}dx+x^y\ln x\,dy=0$.
Explanation of Method
Check exactness with $M=yx^{y-1}$, $N=x^y\ln x$, and recognize the potential function directly as $F=x^y$.
Step 1: Check exactness
Using $x^y=e^{y\ln x}$: $M_y=x^{y-1}(1+y\ln x)=N_x$, so the equation is exact.
Step 2: Find the potential function
$F=\int yx^{y-1}dx=x^y+g(y)$. Matching $F_y=x^y\ln x+g'(y)=N=x^y\ln x$ gives $g'(y)=0$.
Result
Final answer:
\[\boxed{\,x^y=C,\qquad C\in\mathbb{R}}\]