Question

Solve the differential equation \(y\,dx+(x-y^2)\,dy=0\).

Hint:

Rewrite as a linear ODE in x(y): dx/dy + x/y = y, then use integrating factor y.

Answer 1

Step 1: Rearranging as linear in x:
\(y\,dx=-(x-y^2)\,dy \Rightarrow \dfrac{dx}{dy}=-\dfrac{x-y^2}{y}=-\dfrac{x}{y}+y\), i.e. \(\dfrac{dx}{dy}+\dfrac{1}{y}x=y\) — a linear ODE in \(x\) as a function of \(y\).

Step 2: Finding the integrating factor:
I.F. \(=e^{\int \frac1y dy}=e^{\ln y}=y\).

Step 3: Multiplying through and integrating:
\(\dfrac{d}{dy}(x\cdot y)=y\cdot y=y^2\Rightarrow xy=\displaystyle\int y^2\,dy=\dfrac{y^3}{3}+C\).

Final Answer:
\[ \boxed{x=\dfrac{y^2}{3}+\dfrac{C}{y}} \]