Which of the following differential equations has the solution y=3x^2?

(a) \frac{d^2 y}{dx^2}-6x=0

(b) \frac{dy}{dx}-3x=0

(c) x \frac{d^2 y}{dx^2}–\frac{dy}{dx}=0

(d) \frac{d^2 y}{dx^2}-\frac{3dy}{dx}=0

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I want to ask this question from General and Particular Solutions of Differential Equation in chapter Differential Equations of Mathematics – Class 12

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Right answer is (c) x \frac{d^2 y}{dx^2}–\frac{dy}{dx}=0

The explanation is: Consider the function y=3x^2

Differentiating w.r.t x, we get

\frac{dy}{dx}=6x –(1)

Differentiating (1) w.r.t x, we get

\frac{d^2 y}{dx^2}=6

∴\frac{xd^2 y}{dx^2}-\frac{6dy}{dx}=6x-6x=0

Hence, the function y=3x^2 is a solution for the differential equation x \frac{d^2 y}{dx^2}-6 \frac{dy}{dx}=0.