Let $b$ be a nonzero real number. Suppose $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation $f'(x) = \frac{f(x)}{b^{2} + x^{2}}$ for all $x \in \mathbb{R}$, then which of the following statements is/are True?

$f'(x) = \dfrac{f(x)}{b^2+x^2}, \quad f(0)=1, \quad b\neq 0$

$\text{To find: which of statements (1)--(4) are True}$

$\dfrac{df}{f} = \dfrac{dx}{b^2+x^2}$

$\ln f = \dfrac{1}{b}\tan^{-1}\!\left(\dfrac{x}{b}\right) + C$

$x=0,\ f=1:\ 0 = 0 + C \;\Rightarrow\; C=0$

$f(x) = e^{\frac{1}{b}\tan^{-1}(x/b)}$

$f>0,\ b^2+x^2>0 \;\Rightarrow\; f'(x)=\dfrac{f(x)}{b^2+x^2} > 0$

$\Rightarrow (1) \text{ is True}$

$f' > 0 \ \text{even when } b<0 \;\Rightarrow\; (2) \text{ is False}$

$g(x)=\dfrac{1}{b}\tan^{-1}\!\left(\dfrac{x}{b}\right) \text{ is odd} \;\Rightarrow\; f(-x)=e^{-g(x)}=\dfrac{1}{f(x)}$

$\Rightarrow f(x)f(-x)=1 \;\Rightarrow\; (3) \text{ is True}$

$f(x)-f(-x) = e^{g(x)}-e^{-g(x)} = 2\sinh(g(x)) \neq 0$

$\Rightarrow (4) \text{ is False}$

$\boxed{\text{True statements: (1), (3)}}$

$\text{Answer: (1), (3)}$