56.3.31 problem 31

Internal problem ID [8889]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:18:38 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} v v^{\prime }&=\frac {2 v^{2}}{r^{3}}+\frac {\lambda r}{3} \end{align*}

✓ Solution by Maple

Time used: 0.023 (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 \,\operatorname {Ei}_{1}\left (-\frac {2}{r^{2}}\right ) \lambda +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 \,\operatorname {Ei}_{1}\left (-\frac {2}{r^{2}}\right ) \lambda +3 c_{1} \right )}\, {\mathrm e}^{-\frac {2}{r^{2}}}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.745 (sec). Leaf size: 98

DSolve[v[r]*D[v[r],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*}