Let f: (0, ∞) → R be a differentiable function such that f ^ ' (x)...

Let $y = f\left(x\right)$
$\dfrac{dy}{dx} + \dfrac{y}{x} = 2$ (linear first order differential equation)

$I . F . = {e}^{∫\dfrac{dx}{x}} = {e}^{\log_{e}\left(x\right)} = x$, (I.F.=Integrating factor)

$∴ y \cdot x = 2 ∫ xdx + c$
$∴ yx = {x}^{2} + c$

$⇒ f\left(x\right) = x + \dfrac{c}{x};$
As $f\left(1\right) \ne 1 ⇒ c \ne 0$
$⇒ {f}^{'}\left(x\right) = 1 - \dfrac{c}{{x}^{2}}, c \ne 0$
$\lim_{x \to {0}^{+}} {f}^{'}\left(\dfrac{1}{x}\right) = \lim_{x \to {0}^{+}} \left(1 - c{x}^{2}\right) = 1$
$\lim_{x \to {0}^{+}} xf\left(\dfrac{1}{x}\right) = \lim_{x \to {0}^{+}} x\left(\dfrac{1}{x} + cx\right) = \lim_{x \to {0}^{+}} \left(1 + c{x}^{2}\right) = 1$
$\lim_{x \to {0}^{+}} {x}^{2}{f}^{'}\left(x\right) = \lim_{x \to {0}^{+}} {x}^{2}\left(1 - \dfrac{c}{{x}^{2}}\right) = \lim_{x \to {0}^{+}} \left({x}^{2} - c\right) = - c \ne 0$ as $c \ne 0$
$f\left(x\right) = x + \dfrac{c}{x}, c \ne 0$
$∴ \lim_{x \to {0}^{+}} f\left(x\right) = \infty$ $⇒$function is not bounded in \left(0, 2\right)$.