Question

What is the solution of dy/dx = (6x + 9y – 7)/(2x + 3y – 6)?

(a) 3x – y + log|2x + 3y – 3| = -c/3

(b) 3x – y + log|2x + 3y – 3| = c/3

(c) 3x + y + log|2x + 3y – 3| = -c/3

(d) 3x – y –  log|2x + 3y – 3| = c/3

This question was posed to me in an internship interview.

Question is taken from Linear First Order Differential Equations in division Differential Equations of Mathematics – Class 12

Answer 1

The correct answer is (a) 3x – y + log|2x + 3y – 3| = -c/3

Easy explanation: dy/dx = (6x + 9y – 7)/(2x + 3y – 6)

So, dy/dx = (3(2x + 3y) – 7)/(2x + 3x – 6)  ……….(1)

Now, we put, 2x + 3y = z

Therefore, 2 + 3dy/dx = dz/dx [differentiating with respect to x]

Or, dy/dx = 1/3(dz/dx – 2)

Therefore, from (1) we get,

1/3(dz/dx – 2) = (3z – 7)/(z – 6)

Or, dz/dx = 2 + (3(3z – 7))/(z – 6)

= 11(z – 3)/(z – 6)

Or, (z – 6)/(z – 3) dz = 11 dx

Or, ∫(z – 6)/(z – 3) dz = ∫11 dx

Or, ∫(1 – 3/(z – 3)) dz = 11x + c

Or, z – log |z – 3| = 11x + c

Or, 2x + 3y – 11x – 3log|2x + 3y -3| = c

Or, 3y – 9x – 3log|2x + 3y – 3| = c

Or, 3x – y + log|2x + 3y – 3| = -c/3