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
:
Monday, January 27, 2025 at 01:30:44 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} r^{\prime \prime }&=-\frac {k}{r^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.044 (sec). Leaf size: 369
dsolve(diff(r(t),t$2)=-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}-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}-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 )}\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}-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}+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}+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 )}\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}+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: 0.177 (sec). Leaf size: 65
DSolve[D[r[t],{t,2}]==-k/(r[t]^2),r[t],t,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left (\frac {r(t) \sqrt {\frac {2 k}{r(t)}+c_1}}{c_1}-\frac {2 k \text {arctanh}\left (\frac {\sqrt {\frac {2 k}{r(t)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(t+c_2){}^2,r(t)\right ]
\]