If, A and B are arbitrary constants then what will be the differential equation of y = Ax + B/x?

Category: QuestionsIf, A and B are arbitrary constants then what will be the differential equation of y = Ax + B/x?
Editor">Editor Staff asked 11 months ago

If, A and B are arbitrary constants then what will be the differential equation of y = Ax + B/x?
 
(a) x^2 d^2 y/dx^2 – xdy/dx + y = 0
 
(b) x^2 d^2 y/dx^2 + xdy/dx + y = 0
 
(c) x^2 d^2 y/dx^2 + xdy/dx – y = 0
 
(d) x^2 d^2 y/dx^2 – xdy/dx – y = 0
 
I have been asked this question in quiz.
 
My doubt is from Linear Second Order Differential Equations topic in section 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^2 d^2 y/dx^2 + xdy/dx – y = 0
 
Easiest explanation: Given, y = Ax + B/x
 
=> xy = Ax^2 + B  ……….(1)
 
Differentiating (1) with respect to x, we get,
 
d(xy)/dx = d/dx(Ax^2 + B)
 
or, xdy/dx + y = A * 2x  ……….(2)
 
Differentiating again with respect to x, we get,
 
x*d^2y/dx^2 + dy/dx + dy/dx = A*2  ……….(3)
 
Eliminating A from (2) and (3) we get,
 
x^2 d^2 y/dx^2 + 2xdy/dx = 2Ax  [multiplying (3) by x]
 
or, x^2 d^2 y/dx^2 + 2xdy/dx = xdy/dx + y [using (2)]
 
or, x^2 d^2 y/dx^2 + xdy/dx – y = 0