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

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

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
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

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