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
:
Monday, January 27, 2025 at 01:32:02 PM
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*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 441
dsolve(diff(r(t),t$2)= h^2/r(t)^3-k/r(t)^2,r(t), singsol=all)
\begin{align*}
r &= \frac {c_1 \left (c_1^{2} k^{2}-2 k c_1 \,{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}}{2} \\
r &= \frac {c_1 \left (c_1^{2} k^{2}-2 k c_1 \,{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} k^{2}+2 \textit {\_Z} \,c_1^{3} k \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2} h^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) t \right )}}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.091 (sec). Leaf size: 130
DSolve[D[r[t],{t,2}]==h^2/r[t]^3-k/r[t]^2,r[t],t,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {\left (\sqrt {c_1} \left (-h^2+r(t) (2 k+c_1 r(t))\right )-k \sqrt {-h^2+r(t) (2 k+c_1 r(t))} \text {arctanh}\left (\frac {k+c_1 r(t)}{\sqrt {c_1} \sqrt {-h^2+r(t) (2 k+c_1 r(t))}}\right )\right ){}^2}{c_1{}^3 r(t)^2 \left (-\frac {h^2}{r(t)^2}+\frac {2 k}{r(t)}+c_1\right )}=(t+c_2){}^2,r(t)\right ]
\]