Q.Form the differential equation representing the family of curves $y = Ax + Bx^2$, where $A$ and $B$ are arbitrary constants.

Two arbitrary constants require differentiating twice before both can be eliminated.

The family $y=Ax+Bx^2$ has two arbitrary constants, so two differentiations are needed.

First derivative: $\dfrac{dy}{dx} = A + 2Bx$.

Second derivative: $\dfrac{d^2y}{dx^2} = 2B$, so $B = \dfrac12\dfrac{d^2y}{dx^2}$.

Substituting $B$ into the first-derivative equation: $A = \dfrac{dy}{dx} - 2Bx = \dfrac{dy}{dx} - x\dfrac{d^2y}{dx^2}$.

Substituting both $A$ and $B$ into the original relation $y=Ax+Bx^2$:

$y = \left(\dfrac{dy}{dx}-x\dfrac{d^2y}{dx^2}\right)x + \dfrac12\dfrac{d^2y}{dx^2}x^2 = x\dfrac{dy}{dx} - x^2\dfrac{d^2y}{dx^2} + \dfrac12 x^2\dfrac{d^2y}{dx^2} = x\dfrac{dy}{dx} - \dfrac12 x^2\dfrac{d^2y}{dx^2}$ …

Stopping after only ONE differentiation (leaving both $A$ and $B$ present, since one equation cannot isolate two unknowns) is the most common error — a family with two arbitrary constants always needs two differentiation steps …