What will be the value of dy/dx – a/x * y = (x + 1)/x?

Category: QuestionsWhat will be the value of dy/dx – a/x * y = (x + 1)/x?
Editor">Editor Staff asked 11 months ago

What will be the value of dy/dx – a/x * y = (x + 1)/x?
 
(a) y = x/(1 – a) – 1/a + cx^a
 
(b) y = x/(1 + a) + 1/a + cx^a
 
(c) y = x/(1 – a) – 1/a – cx^a
 
(d) y = x/(1 + a) – 1/a + cx^a
 
I got this question in examination.
 
The doubt is from Linear First Order Differential Equations topic in chapter 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

Correct option is (a) y = x/(1 – a) – 1/a + cx^a
 
The best explanation: dy/dx – a/x * y = (x + 1)/x …….(1)
 
Multiplying both sides of equation (1) by
 
e^∫-a/xdx
 
= e^-a log x
 
= e^log x^-a
 
= x^-a
 
We get, x^-ady/dx – x^-a (a/x)y = x^-a (x + 1)/x
 
Or, d/dx(y . x^-a) = x^-a + x^-a – 1 …….(2)
 
Integrating both sides of (2) we get,
 
y. x^-a = x^-a + 1/(-a + 1) + x^-a – 1 + 1/(-a -1 + 1) + c
 
= x^-a.x/(1 – a) + x^-a/-a + c
 
Or, y = x/(1 – a) – 1/a + cx^a