2.1.15.7 ✓ Mathematica. Time used: 0.486 (sec). Leaf size: 33

ode=D[x[t],t]==Cos[x[t]/t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{gather*} \text {Solve}\left [\int _1^{\frac {x(t)}{t}}\frac {1}{K[1]-\cos (K[1])}dK[1]=-\log (t)+c_1,x(t)\right ] \end{gather*}