Question

What will be the value of dy/dx = (x + 2y + 3)/(2x + 3y + 4)?

(a) [(2 + √3)/2√3 * (log (√3(y + 2) – (x – 1))) + (2 – √3)/2√3 * (log (√3(y + 2) – (x – 1)))]

(b) [(2 + √3)/2√3 * (log (√3(y + 2) + (x – 1))) – (2 – √3)/2√3 * (log (√3(y + 2) + (x – 1)))]

(c) [(2 + √3)/2√3 * (log (√3(y + 2) – (x – 1))) – (2 – √3)/2√3 * (log (√3(y + 2) – (x – 1)))]

(d) [(2 + √3)/2√3 * (log (√3(y + 2) + (x – 1))) – (2 – √3)/2√3 * (log (√3(y + 2) – (x – 1)))]

This question was addressed to me in an interview for internship.

I would like to ask this question from Linear First Order Differential Equations topic in division Differential Equations of Mathematics – Class 12

Answer 1

Correct choice is (c) [(2 + √3)/2√3 * (log (√3(y + 2) – (x – 1))) – (2 – √3)/2√3 * (log (√3(y + 2) – (x – 1)))]

Best explanation: Put x = X + h, Y = Y + k,

We have, dY/dX = (X + 2Y +(h + 2k + 3))/ 2X + 3Y + (2h + 3k + 4)

So, (a – b)x = (a – b)

To determine h and k we set,

2h + 3k + 4 = 0 and h + 2k + 3 = 0

=> h = 1 and k = – 2

Therefore, dY/dX = (X + 2Y) / (2X + 3Y)

Putting Y = VX, we get,

V + X dV/dX = (1 + 2V)/(2 + 3V)

        = (1 + 2V)/(3V^2 – 1)*dV = -dX/X

=> [(2 + √3)/(2(√3V – 1)) – (2 – √3)/(2(√3V – 1))] dV = -dX/X

Simplifying it further, we get;

[(2 + √3)/2√3 * (log (√3Y – X)) – (2 – √3)/2√3 * (log (√3Y – X))]

Where, X = x – 1 and Y = y + 2