Existence and uniqueness analysis

\[\begin {aligned} x^{\prime }&=\frac {1}{\sqrt {t^{2}+1}}\\ x \left (1\right ) &= 0\\ \end {aligned}\]

This is a linear ODE. In canonical form it is written as

\begin{align*} x^{\prime } + q(t)x &= p(t) \end{align*}

Comparing the above to the given ODE shows that

\begin{align*} q(t) &=0\\ p(t) &=\frac {1}{\sqrt {t^{2}+1}} \end{align*}

The domain of \(q(t)=0\) is

\begin{align*} \{-\infty <t <\infty \} \end{align*}

The point \(t_0 = 1\) is inside this domain. The domain of \(p(t)=\frac {1}{\sqrt {t^{2}+1}}\) is

\begin{align*} \{-\infty <t <\infty \} \end{align*}

Because the point \(t_0 = 1\) is inside this domain, solution exists and is unique.