Question

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

Answer 1

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