56.3.31 problem 31
Internal
problem
ID
[8889]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
31
Date
solved
:
Sunday, March 30, 2025 at 01:52:28 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: 8.745 (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: 1.667 (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)
\[
\left [ v{\left (r \right )} = - \frac {\sqrt {3} \sqrt {C_{1} e^{- \frac {2}{r^{2}}} + \lambda _{} r^{2} - 2 \lambda _{} e^{- \frac {2}{r^{2}}} \operatorname {Ei}{\left (\frac {2}{r^{2}} \right )}}}{3}, \ v{\left (r \right )} = \frac {\sqrt {3} \sqrt {C_{1} e^{- \frac {2}{r^{2}}} + \lambda _{} r^{2} - 2 \lambda _{} e^{- \frac {2}{r^{2}}} \operatorname {Ei}{\left (\frac {2}{r^{2}} \right )}}}{3}\right ]
\]