Internal problem ID [7221]
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]
\[ \boxed {v v^{\prime }-\frac {2 v^{2}}{r^{3}}=\frac {\lambda r}{3}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 97
dsolve(v(r)*diff(v(r),r)=2*v(r)^2/r^3+1/3*lambda*r,v(r), singsol=all)
\begin{align*} v \left (r \right ) &= -\frac {\sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )+3 c_{1} \right )}\, {\mathrm e}^{-\frac {2}{r^{2}}}}{3} \\ v \left (r \right ) &= \frac {\sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )+3 c_{1} \right )}\, {\mathrm e}^{-\frac {2}{r^{2}}}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.758 (sec). Leaf size: 98
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 {e^{-\frac {2}{r^2}} \left (-2 \lambda \operatorname {ExpIntegralEi}\left (\frac {2}{r^2}\right )+\lambda e^{\frac {2}{r^2}} r^2+3 c_1\right )}}{\sqrt {3}} \\ v(r)\to \frac {\sqrt {e^{-\frac {2}{r^2}} \left (-2 \lambda \operatorname {ExpIntegralEi}\left (\frac {2}{r^2}\right )+\lambda e^{\frac {2}{r^2}} r^2+3 c_1\right )}}{\sqrt {3}} \\ \end{align*}