2.1.4.2 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

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

Because the ODE has the form \(x^{\prime }=f(t)\), the solution requires only integration. Therefore

\begin{align*} dx &= \left (\frac {1}{\sqrt {t^{2}+1}}\right ) \, dt\\ x &= \int { \left (\frac {1}{\sqrt {t^{2}+1}}\right ) \, dt}\\ &= \operatorname {arcsinh}\left (t \right )+c_1 \end{align*}

Applying the initial condition \(x \left (1\right ) = 0\), the solution becomes

\begin{align*} x &= \operatorname {arcsinh}\left (t \right )-\ln \left (1+\sqrt {2}\right ) \\ \end{align*}

\( x = \operatorname {arcsinh}\left (t \right )-\ln \left (1+\sqrt {2}\right ) \)

Summary of solutions found

\[ x = \operatorname {arcsinh}\left (t \right )-\ln \left (1+\sqrt {2}\right ) \]