Question

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

This 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 1

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.