32.10.11 problem Exercise 35.11, page 504

Internal problem ID [6005]
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 35.11, page 504
Date solved : Wednesday, March 05, 2025 at 12:03:44 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} r^{\prime \prime }&=\frac {h^{2}}{r^{3}}-\frac {k}{r^{2}} \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 441

ode:=diff(diff(r(t),t),t) = h^2/r(t)^3-k/r(t)^2; 
dsolve(ode,r(t), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.091 (sec). Leaf size: 130

ode=D[r[t],{t,2}]==h^2/r[t]^3-k/r[t]^2; 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\left (\sqrt {c_1} \left (-h^2+r(t) (2 k+c_1 r(t))\right )-k \sqrt {-h^2+r(t) (2 k+c_1 r(t))} \text {arctanh}\left (\frac {k+c_1 r(t)}{\sqrt {c_1} \sqrt {-h^2+r(t) (2 k+c_1 r(t))}}\right )\right ){}^2}{c_1{}^3 r(t)^2 \left (-\frac {h^2}{r(t)^2}+\frac {2 k}{r(t)}+c_1\right )}=(t+c_2){}^2,r(t)\right ] \]

✓ Sympy. Time used: 54.732 (sec). Leaf size: 48

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

\[ \left [ \int \limits ^{r{\left (t \right )}} \frac {1}{\sqrt {C_{1} - \frac {- 2 u k + h^{2}}{u^{2}}}}\, du = C_{2} + t, \ \int \limits ^{r{\left (t \right )}} \frac {1}{\sqrt {C_{1} - \frac {- 2 u k + h^{2}}{u^{2}}}}\, du = C_{2} - t\right ] \]