32.10.7 problem Exercise 35.7, page 504
Internal
problem
ID
[6001]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
8.
Special
second
order
equations.
Lesson
35.
Independent
variable
x
absent
Problem
number
:
Exercise
35.7,
page
504
Date
solved
:
Wednesday, March 05, 2025 at 12:02:24 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} r^{\prime \prime }&=-\frac {k}{r^{2}} \end{align*}
✓ Maple. Time used: 0.044 (sec). Leaf size: 369
ode:=diff(diff(r(t),t),t) = -k/r(t)^2;
dsolve(ode,r(t), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.177 (sec). Leaf size: 65
ode=D[r[t],{t,2}]==-k/(r[t]^2);
ic={};
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {r(t) \sqrt {\frac {2 k}{r(t)}+c_1}}{c_1}-\frac {2 k \text {arctanh}\left (\frac {\sqrt {\frac {2 k}{r(t)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(t+c_2){}^2,r(t)\right ]
\]
✓ Sympy. Time used: 6.018 (sec). Leaf size: 201
from sympy import *
t = symbols("t")
k = symbols("k")
r = Function("r")
ode = Eq(k/r(t)**2 + Derivative(r(t), (t, 2)),0)
ics = {}
dsolve(ode,func=r(t),ics=ics)
\[
\left [ - t + \frac {\sqrt {2} r^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} r{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {r{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} r{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {r{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}, \ t + \frac {\sqrt {2} r^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} r{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {r{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} r{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {r{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}\right ]
\]