50.3.31 problem 31

Internal problem ID [10175]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 07:11:05 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 97

ode:=v(r)*diff(v(r),r) = 2*v(r)^2/r^3+1/3*lambda*r; 
dsolve(ode,v(r), singsol=all);
 

\begin{align*} v &= -\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 &= \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*}

✓ Mathematica. Time used: 8.116 (sec). Leaf size: 98

ode=v[r]*D[v[r],r]==2*v[r]^2/r^3+1/3*\[Lambda]*r; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.083 (sec). Leaf size: 87

from sympy import * 
r = symbols("r") 
lambda_ = symbols("lambda_") 
v = Function("v") 
ode = Eq(-lambda_*r/3 + v(r)*Derivative(v(r), r) - 2*v(r)**2/r**3,0) 
ics = {} 
dsolve(ode,func=v(r),ics=ics)
 

\[ \left [ v{\left (r \right )} = - \frac {\sqrt {3} \sqrt {C_{1} e^{- \frac {2}{r^{2}}} + \lambda _{} r^{2} - 2 \lambda _{} e^{- \frac {2}{r^{2}}} \operatorname {Ei}{\left (\frac {2}{r^{2}} \right )}}}{3}, \ v{\left (r \right )} = \frac {\sqrt {3} \sqrt {C_{1} e^{- \frac {2}{r^{2}}} + \lambda _{} r^{2} - 2 \lambda _{} e^{- \frac {2}{r^{2}}} \operatorname {Ei}{\left (\frac {2}{r^{2}} \right )}}}{3}\right ] \]