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

Category: QuestionsWhich of the following differential equations has the solution y=3x^2?
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

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.