Q.Solve the differential equation $\dfrac{dy}{dx} = \dfrac{x^2}{y^2}$ by the method of variable separation.

The right side is a function of $x$ divided by a function of $y$, so the variables separate directly.

The equation $\dfrac{dy}{dx}=\dfrac{x^2}{y^2}$ has its right side already in the form (function of $x$) ÷ (function of $y$), so it is variable-separable.

Separating: $y^2\,dy = x^2\,dx$.

Integrating both sides independently, using $\int x^n\,dx = \dfrac{x^{n+1}}{n+1}+C$: $\dfrac{y^3}{3} = \dfrac{x^3}{3} + c$.

Multiplying through by 3 and combining the constants into one: $y^3 - x^3 = k$ (where $k=3c$), the general solution. …

A common error is integrating $y^2\,dy$ into $\dfrac{y^3}{3}$ correctly but then forgetting to also multiply that term by 3 when clearing fractions at the end — always carry the same multiplying step through EVERY term, includin …