Question

Show that y = ax + (b/x), x ≠ 0, is a solution of the differential equation x²y”+ xy’ – y = 0

Answer 1

Given solution: y = ax + (b/x), x ≠ 0

Here ‘a’ and ‘b’ are arbitrary constants

y = ax + (b/x) ... (1) 

xy = ax² + b 

Differentiate with respect to ‘x’ 

xy’ + y . 1 = a (2x) = 2ax ... (2) 

Differentiate again with respect to ‘x’. 

xy” + y’ . 1 + y = 2a ⇒ xy” + 2y’ = 2a … (3) 

Substitute (3) in (2) 

xy’ + y = (xy” + 2y’)x 

xy’ + y = x²y” + 2xy’ ⇒ x²y” + xy’ – y = 0 

Hence proved.