2.6.12.5 ✓ Maple. Time used: 0.001 (sec). Leaf size: 14
ode:=diff(v(u),u)+2*u*v(u) = 2*u;
dsolve(ode,v(u), singsol=all);
\begin{align*} v = 1+{\mathrm e}^{-u^{2}} c_1 \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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )+2 u v \left (u \right )=2 u \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d u}v \left (u \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )=-2 u v \left (u \right )+2 u \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d u}v \left (u \right )}{v \left (u \right )-1}=-2 u \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} u \\ {} & {} & \int \frac {\frac {d}{d u}v \left (u \right )}{v \left (u \right )-1}d u =\int -2 u d u +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (v \left (u \right )-1\right )=-u^{2}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (u \right ) \\ {} & {} & v \left (u \right )={\mathrm e}^{-u^{2}+\mathit {C1}}+1 \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & v \left (u \right )=\mathit {C1} \,{\mathrm e}^{-u^{2}}+1 \end {array} \]