2.6.2.3 ✓ Maple. Time used: 0.001 (sec). Leaf size: 13

ode:=sec(theta)^2 = m/k*diff(s(theta),theta); 
dsolve(ode,s(theta), singsol=all);
 

\begin{align*} s = \frac {k \tan \left (\theta \right )}{m}+c_1 \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sec \left (\theta \right )^{2}=\frac {m \left (\frac {d}{d \theta }s \left (\theta \right )\right )}{k} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d \theta }s \left (\theta \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {d}{d \theta }s \left (\theta \right )=\frac {\sec \left (\theta \right )^{2} k}{m} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} \theta \\ {} & {} & \int \left (\frac {d}{d \theta }s \left (\theta \right )\right )d \theta =\int \frac {\sec \left (\theta \right )^{2} k}{m}d \theta +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & s \left (\theta \right )=\frac {k \tan \left (\theta \right )}{m}+\mathit {C1} \end {array} \]