2.1.15.6 ✓ Maple. Time used: 0.004 (sec). Leaf size: 27

ode:=diff(x(t),t) = cos(x(t)/t); 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {1}{-\cos \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +\ln \left (t \right )+c_1 \right ) t \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying homogeneous D 
<- homogeneous successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=\cos \left (\frac {x \left (t \right )}{t}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=\cos \left (\frac {x \left (t \right )}{t}\right ) \end {array} \]