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: Recognising it can't be separated directly:
Written as \(\dfrac{dx}{dy}=y-\dfrac{x}{y}\), the \(x\) and \(y\) terms are entangled, confirming this is genuinely a first-order linear equation in \(x(y)\), not separable in \(x,y\) directly.

Step 2: Standard linear form:
Compare with \(\dfrac{dx}{dy}+P(y)x=Q(y)\): here \(P(y)=\dfrac1y\), \(Q(y)=y\).

Step 3: Integrating factor and solution formula:
I.F.\(=e^{\int P\,dy}=y\); general solution \(x\cdot(\text{I.F.})=\displaystyle\int Q\cdot(\text{I.F.})\,dy+C\), i.e. \(xy=\int y\cdot y\,dy+C=\dfrac{y^3}{3}+C\).

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