Question

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

(a) a^2 log (x + √(a^2 – x^2)) + c

(b) a^2 log (x + √( a^2 + x^2)) + c

(c) a^2 log (x – √( a^2 + x^2)) + c

(d) a^2 log (x – √( a^2 – x^2)) + c

The question was asked in an interview for internship.

This intriguing question originated from Linear First Order Differential Equations topic in division Differential Equations of Mathematics – Class 12

Answer 1

The correct answer is (b) a^2 log (x + √( a^2 + x^2)) + c

To elaborate: The given form of equation can be written as,

dy/dx + 1/√(a^2 + x^2) * y = (√(a^2 + x^2) – x)/√(a^2 + x^2) ……(1)

We have, ∫1/√(a^2 + x^2)dx = log(x + √(a^2 + x^2))

Therefore, integrating factor is,

e^∫1/√(a^2 + x^2) = e^log(x + √(a^2 + x^2))

= x + √(a^2 + x^2)

Therefore, multiplying both sides of (1) by x + √(a^2 + x^2) we get,

x + √(a^2 + x^2dy/dx + (x + √(a^2 + x^2))/ √(a^2 + x^2)*y = (x + √(a^2 + x^2))(√(a^2 + x^2) – x)/√(a^2 + x^2)

or, d/dx[x + √(a^2 + x^2)*y] = (a^2 + x^2) ………..(2)

Integrating both sides of (2) we get,

(x + √(a^2 + x^2) * y = a^2∫dx/√(a^2 + x^2)

= a^2 log (x + √(a^2 + x^2)) + c