38.1.6 problem 6
Internal
problem
ID
[8167]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
1.
Introduction
to
differential
equations.
Exercises
1.1
at
page
12
Problem
number
:
6
Date
solved
:
Tuesday, September 30, 2025 at 05:16:25 PM
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: 0.059 (sec). Leaf size: 257
ode:=diff(diff(R(t),t),t) = -k/R(t)^2;
dsolve(ode,R(t), singsol=all);
\begin{align*}
R &= \frac {c_1 \left ({\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 )} c_1^{2} k^{2}+{\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 k c_1 \right )}{2} \\
R &= \frac {c_1 \left ({\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 )} c_1^{2} k^{2}+{\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 k c_1 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.099 (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: 4.241 (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)
\[
\left [ - t + \frac {\sqrt {2} R^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {R{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {R{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}, \ t + \frac {\sqrt {2} R^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {R{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {R{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}\right ]
\]