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

Separate the variables, integrate using the power rule for negative exponents, and simplify.

The equation $\dfrac{dy}{dx}=\dfrac{y^2}{x^2}$ separates as $\dfrac{dy}{y^2} = \dfrac{dx}{x^2}$, i.e. $y^{-2}\,dy = x^{-2}\,dx$.

Integrating both sides using $\int t^{-2}\,dt = -\dfrac1t+C$: $-\dfrac1y = -\dfrac1x + c$.

Rearranging: $\dfrac1x - \dfrac1y = c$ — the general solution, expressed as a relation between $x$ and $y$. …

A common error when integrating a negative power like $y^{-2}$ is to forget the resulting negative sign, writing $\int y^{-2}dy = \dfrac1y$ instead of the correct $-\dfrac1y$ — always apply the power rule $\int t^n …