Internal problem ID [6468]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
Solve \begin {gather*} \boxed {v v^{\prime }-\frac {2 v^{2}}{r^{3}}-\frac {\lambda r}{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 101
dsolve(v(r)*diff(v(r),r)=2*v(r)^2/r^3+1/3*lambda*r,v(r), singsol=all)
\begin{align*} v \relax (r ) = -\frac {{\mathrm e}^{-\frac {2}{r^{2}}} \sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \expIntegral \left (1, -\frac {2}{r^{2}}\right ) \lambda +3 c_{1}\right )}}{3} \\ v \relax (r ) = \frac {{\mathrm e}^{-\frac {2}{r^{2}}} \sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \expIntegral \left (1, -\frac {2}{r^{2}}\right ) \lambda +3 c_{1}\right )}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.691 (sec). Leaf size: 86
DSolve[v[r]*v'[r]==2*v[r]^2/r^3+1/3*\[Lambda]*r,v[r],r,IncludeSingularSolutions -> True]
\begin{align*} v(r)\to -\frac {\sqrt {\lambda r^2+e^{-\frac {2}{r^2}} \left (-2 \lambda \text {Ei}\left (\frac {2}{r^2}\right )+3 c_1\right )}}{\sqrt {3}} \\ v(r)\to \frac {\sqrt {\lambda r^2+e^{-\frac {2}{r^2}} \left (-2 \lambda \text {Ei}\left (\frac {2}{r^2}\right )+3 c_1\right )}}{\sqrt {3}} \\ \end{align*}