If y = t(x) be a differentiable function ᵾ x € R, then which of the following is always true?

Category: QuestionsIf y = t(x) be a differentiable function ᵾ x € R, then which of the following is always true?
Editor">Editor Staff asked 11 months ago

If y = t(x) be a differentiable function ᵾ x € R, then which of the following is always true?
 
(a) d^2y/dx^2 – (dy/dx)^3 = 0
 
(b) d^2y/dx^2 + (dy/dx)^3 d^2x/dy^2 = 0
 
(c) d^2y/dx^2 – (dx/dy)^3 = 0
 
(d) d^2y/dx^2 + (dy/dx)^3 = 0
 
I had been asked this question during a job interview.
 
My question is from Linear Second Order Differential Equations topic in portion 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

Correct option is (b) d^2y/dx^2 + (dy/dx)^3 d^2x/dy^2 = 0
 
For explanation I would say: (dy/dx)= (dx/dy)^-1
 
So, d^2y/dx^2 = – (dx/dy)^-2 d/dx(dx/dy)
 
= – (dy/dx)^2(d^2x/dy^2)(dy/dx)
 
=> d^2y/dx^2 + (dy/dx)^3 d^2x/dy^2 = 0