2.1.20.6 ✓ Maple. Time used: 0.000 (sec). Leaf size: 8

ode:=diff(x(t),t)+x(t)*tan(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x = c_1 \cos \left (t \right ) \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )+x \left (t \right ) \tan \left (t \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=-x \left (t \right ) \tan \left (t \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d t}x \left (t \right )}{x \left (t \right )}=-\tan \left (t \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\frac {d}{d t}x \left (t \right )}{x \left (t \right )}d t =\int -\tan \left (t \right )d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (x \left (t \right )\right )=\ln \left (\cos \left (t \right )\right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \left (t \right ) \\ {} & {} & x \left (t \right )={\mathrm e}^{\mathit {C1}} \cos \left (t \right ) \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & x \left (t \right )=\mathit {C1} \cos \left (t \right ) \end {array} \]