Problem 115
Find the general integral of the total differential equation…
Find the general integral of the total differential equation $\left(4-\dfrac{y^2}{x^2}\right)dx+2\dfrac{y}{x}dy=0$.
Explanation of Method
Check exactness with $M=4-y^2/x^2$, $N=2y/x$, and integrate.
Step 1: Check exactness
$M_y=-\dfrac{2y}{x^2}=N_x$, so the equation is exact.
Step 2: Find the potential function
$F=\int\!\left(4-\dfrac{y^2}{x^2}\right)dx=4x+\dfrac{y^2}{x}+g(y)$. Matching $F_y=\dfrac{2y}{x}+g'(y)=N$ gives $g'(y)=0$.
Result
Final answer:
\[\boxed{\,4x^2+y^2=Cx,\qquad C\in\mathbb{R}}\]
a family of circles/ellipses through the origin.