What will be the value of the differential equation dy/dx = (x + y)^2/(x + 2)(y – 2)?

Category: QuestionsWhat will be the value of the differential equation dy/dx = (x + y)^2/(x + 2)(y – 2)?
Editor">Editor Staff asked 11 months ago

What will be the value of the differential equation dy/dx = (x + y)^2/(x + 2)(y – 2)?
 
(a) ke^Y/X where, X = x + 2 and Y = y – 2
 
(b) ke^2Y/X where, X = x + 2 and Y = y – 2
 
(c) ke^2Y/X where, X = x and Y = 2y
 
(d) ke^2X/Y where, X = x – 2 and Y = y + 2
 
The question was posed to me during a job interview.
 
This is a very interesting question from Linear First 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) ke^2Y/X where, X = x + 2 and Y = y – 2
 
For explanation I would say: Let, X = x + 2 and Y = y – 2
 
Then, dY/dX = (X + Y)^2/XY
 
So, let, Y = vX
 
dY/dX = v + x dv/dX
 
=>v + Xdv/dX = (v + 1)^2/v
 
=>v/1 + 2v dv = dX/X
 
=> (1 – 1/1 + 2v)dv = 2dx/x
 
=> v – ½ log (1 + 2v) = 2 log X + c
 
This means, X^4(1 + 2Y/X)
 
= ke^2Y/X where, X = x +2 and Y = y – 2