y = c1.x + c2/x ------- (1)
y' = c1 - c2/x^2 ------- (2)
y'' = 2c2/x^3 ------------ (3)
Rearrangng the above eqns,
x^3y'' = 2c2 -------------- (4)
xy' = c1.x - c2.x^(-1) --- (5)
y = c1.x + c2.x^(-1) ---- (6)
we want to get rid of the constants, c1 and c2, so subract (6) from (5),
xy' - y = -2c2.x^(-1)
xy' - y = -(x^3.y'')x^(-1) using (4)
xy' - y = -x^2.y''
x^2.y'' + xy' - y = 0