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Question:

Let f:(0,∞)→R be a differentiable function such that f'(x) = 2 − f(x)/x for all x∈(0,∞) and f(1)≠1. Then:

limx→0+x²f'(x)=0

limx→0+f'(1/x)=1

|f(x)|≤2 for all x∈(0,2)

limx→0+xf(1/x)=2

Solution:

f'(x) = 2 − f(x)/x or dy/dx + y/x = 2
Integrating factor = e∫1/x dx = elnx = x
Solution is yx = ∫2x dx = x² + c or y = x + cx c ≠ 0 as f(1) ≠ 1
(A) limx→0+f'(1/x) = limx→0+(1 − cx²) = 1
(B) limx→0+xf(1/x) = limx→0+1 + cx² = 1
(C) limx→0+x²f'(x) = limx→0+x² − c = −c ≠ 0
(D) limx→0+f(x) → +∞ or −∞