2.6.2.1 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

\[\begin {aligned} \sec \left (\theta \right )^{2}&=\frac {m s^{\prime }}{k} \end {aligned}\]

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

\begin{align*} ds &= \left (\frac {\sec \left (\theta \right )^{2} k}{m}\right ) \, d\theta \\ s &= \int { \left (\frac {\sec \left (\theta \right )^{2} k}{m}\right ) \, d\theta }\\ &= \frac {k \tan \left (\theta \right )}{m}+c_1 \end{align*}

Summary of solutions found

\[ s = \frac {k \tan \left (\theta \right )}{m}+c_1 \]