Question

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

Answer 1

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