44.1.6 problem 6

Internal problem ID [6881]
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 : Wednesday, March 05, 2025 at 02:47:40 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: 0.017 (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: 0.172 (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: 6.920 (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)
 

\[ \left [ - t + \frac {\sqrt {2} R^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {R{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {R{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}, \ t + \frac {\sqrt {2} R^{\frac {3}{2}}{\left (t \right )}}{2 \sqrt {k} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} + \frac {\sqrt {2} \sqrt {k} \sqrt {R{\left (t \right )}}}{C_{1} \sqrt {\frac {C_{1} R{\left (t \right )}}{2 k} + 1}} - \frac {2 k \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {R{\left (t \right )}}}{2 \sqrt {k}} \right )}}{C_{1}^{\frac {3}{2}}} = C_{2}\right ] \]