Find the differential equation for the family of curves given by $y = ax + \dfrac{b}{x}$ where a and b are arbitrary constants.
The given equation is $y = ax + \dfrac{b}{x}$
i.e. $xy = ax^2 + b$ $\;\;\; \cdots \; (1)$
Differentiating equation $(1)$ w.r.t x gives
$x \; \dfrac{dy}{dx} + y = 2ax$ $\;\;\; \cdots \; (2)$
Differentiating equation $(2)$ w.r.t x gives
$x \; \dfrac{d^2 y}{dx^2} + \dfrac{dy}{dx} + \dfrac{dy}{dx} = 2a$
i.e. $x \; \dfrac{d^2 y}{dx^2} + 2 \; \dfrac{dy}{dx} = 2a$ $\;\;\; \cdots \; (3)$
Substituting the value of $2a$ from equation $(3)$ in equation $(2)$ gives
$x \; \dfrac{dy}{dx} + y = x \left(x \; \dfrac{d^2 y}{dx^2} + 2 \; \dfrac{dy}{dx}\right)$
i.e. $x \; \dfrac{dy}{dx} + y = x^2 \; \dfrac{d^2 y}{dx^2} + 2 \; x \; \dfrac{dy}{dx}$
i.e. $x^2 \; \dfrac{d^2 y}{dx^2} + x \; \dfrac{dy}{dx} - y = 0$
is the required differential equation.