2.6.7.6 ✓ Maple. Time used: 0.002 (sec). Leaf size: 13

ode:=diff(v(u),u)+2*v(u)/u = 3*v(u); 
dsolve(ode,v(u), singsol=all);
 

\begin{align*} v = \frac {c_1 \,{\mathrm e}^{3 u}}{u^{2}} \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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )+\frac {2 v \left (u \right )}{u}=3 v \left (u \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d u}v \left (u \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )=-\frac {2 v \left (u \right )}{u}+3 v \left (u \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d u}v \left (u \right )}{v \left (u \right )}=\frac {3 u -2}{u} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} u \\ {} & {} & \int \frac {\frac {d}{d u}v \left (u \right )}{v \left (u \right )}d u =\int \frac {3 u -2}{u}d u +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (v \left (u \right )\right )=3 u -2 \ln \left (u \right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (u \right ) \\ {} & {} & v \left (u \right )=\frac {{\mathrm e}^{3 u +\mathit {C1} -4 \,\mathrm {I} \pi \_Z1}}{u^{2}} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & v \left (u \right )=\frac {\mathit {C1} \,{\mathrm e}^{3 u}}{u^{2}} \end {array} \]