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) \(\frac{d^2 y}{dx^2}-\frac{3dy}{dx}\)=0This question was addressed to me in an international level competition.I want to ask this question from General and Particular Solutions of Differential Equation in chapter Differential Equations of Mathematics – Class 12

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

1.	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) \(\frac{d^2 y}{dx^2}-\frac{3dy}{dx}\)=0This question was addressed to me in an international level competition.I want to ask this question from General and Particular Solutions of Differential Equation in chapter Differential Equations of Mathematics – Class 12
Answer» 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.

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