2.6.13.5 ✓ Maple. Time used: 0.005 (sec). Leaf size: 31
ode:=1+v(u)^2+(u^2+1)*v(u)*diff(v(u),u) = 0;
dsolve(ode,v(u), singsol=all);
\begin{gather*} \begin {aligned} v &= \sqrt {{\mathrm e}^{-2 \arctan \left (u \right )} c_1 -1}\\ v &= -\sqrt {{\mathrm e}^{-2 \arctan \left (u \right )} c_1 -1}\\ \end {aligned} \end{gather*}
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
<- Bernoulli successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 1+v \left (u \right )^{2}+\left (u^{2}+1\right ) v \left (u \right ) \left (\frac {d}{d u}v \left (u \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d u}v \left (u \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )=\frac {-1-v \left (u \right )^{2}}{\left (u^{2}+1\right ) v \left (u \right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (\frac {d}{d u}v \left (u \right )\right ) v \left (u \right )}{-1-v \left (u \right )^{2}}=\frac {1}{u^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} u \\ {} & {} & \int \frac {\left (\frac {d}{d u}v \left (u \right )\right ) v \left (u \right )}{-1-v \left (u \right )^{2}}d u =\int \frac {1}{u^{2}+1}d u +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (v \left (u \right )^{2}+1\right )}{2}=\arctan \left (u \right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (u \right ) \\ {} & {} & \left \{v \left (u \right )=\sqrt {-1+{\mathrm e}^{-2 \arctan \left (u \right )-2 \mathit {C1}}}, v \left (u \right )=-\sqrt {-1+{\mathrm e}^{-2 \arctan \left (u \right )-2 \mathit {C1}}}\right \} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & \left \{v \left (u \right )=\sqrt {-1+\mathit {C1} \,{\mathrm e}^{-2 \arctan \left (u \right )}}, v \left (u \right )=-\sqrt {-1+\mathit {C1} \,{\mathrm e}^{-2 \arctan \left (u \right )}}\right \} \end {array} \]