Q.The differential equation of $y=c^2+\dfrac{c}{x}$ is... (A) $x^4\left(\dfrac{dy}{dx}\right)^2-x\dfrac{dy}{dx}=y$ (B) $\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$ (C) $x^3\left(\dfrac{dy}{dx}\right)^2+x\dfrac{dy}{dx}=y$ (D) $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-y=0

Given a family of curves or relations written with one or more arbitrary constants (for example y = Ae^(3x) + Be^(-3x), or a circle family with a free radius), a differential equation satisfied by every member of that family is formed by differentiating the relation once for each arbitrary constant present, and then eliminating the constants algebraically between the original relation and its derivatives. With one constant, a single differentiation usually already isolates the constant so it can be substituted straight back into the original relation. With two constants, two differentiations are needed, and the elimination typically proceeds by solving one of the derived equations for one constant (or a combination of both), substituting into the other, and simplifying until no constant remains — leaving a relation purely in x, y, and the derivatives up to the order that matches the number of constants eliminated. This is the reverse process to solving a differential equation: instead of starting from an equation and finding its family of solution curves, formation starts from the family and recovers the equation that generated it, and it is also how many geometric word problems (all circles through the origin with centres on an axis, all lines parallel to a given line, and so on) are converted into a solvable differential equation in the first place.