2.6.8.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 32
ode:=(-u^2+1)^(1/2)*diff(v(u),u) = 2*u*(1-v(u)^2)^(1/2);
dsolve(ode,v(u), singsol=all);
\begin{align*} v = \sin \left (\frac {2 c_1 \sqrt {-u^{2}+1}+2 u^{2}-2}{\sqrt {-u^{2}+1}}\right ) \end{align*}
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
<- separable successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sqrt {-u^{2}+1}\, \left (\frac {d}{d u}v \left (u \right )\right )=2 u \sqrt {1-v \left (u \right )^{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d u}v \left (u \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )=\frac {2 u \sqrt {1-v \left (u \right )^{2}}}{\sqrt {-u^{2}+1}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d u}v \left (u \right )}{\sqrt {1-v \left (u \right )^{2}}}=\frac {2 u}{\sqrt {-u^{2}+1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} u \\ {} & {} & \int \frac {\frac {d}{d u}v \left (u \right )}{\sqrt {1-v \left (u \right )^{2}}}d u =\int \frac {2 u}{\sqrt {-u^{2}+1}}d u +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \arcsin \left (v \left (u \right )\right )=\frac {2 \left (u -1\right ) \left (u +1\right )}{\sqrt {-u^{2}+1}}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (u \right ) \\ {} & {} & v \left (u \right )=\sin \left (\frac {\mathit {C1} \sqrt {-u^{2}+1}+2 u^{2}-2}{\sqrt {-u^{2}+1}}\right ) \end {array} \]