15.18.5 problem 5
Internal
problem
ID
[3248]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
35,
page
157
Problem
number
:
5
Date
solved
:
Tuesday, March 04, 2025 at 04:09:08 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} x^{\prime \prime }&=\frac {k^{2}}{x^{2}} \end{align*}
✓ Maple. Time used: 0.710 (sec). Leaf size: 385
ode:=diff(diff(x(t),t),t) = k^2/x(t)^2;
dsolve(ode,x(t), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.191 (sec). Leaf size: 71
ode=D[x[t],{t,2}]==k^2/x[t]^2;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {2 k^2 \text {arctanh}\left (\frac {\sqrt {-\frac {2 k^2}{x(t)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {x(t) \sqrt {-\frac {2 k^2}{x(t)}+c_1}}{c_1}\right ){}^2=(t+c_2){}^2,x(t)\right ]
\]
✓ Sympy. Time used: 2.151 (sec). Leaf size: 405
from sympy import *
t = symbols("t")
k = symbols("k")
x = Function("x")
ode = Eq(-k**2/x(t)**2 + Derivative(x(t), (t, 2)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
\left [ \begin {cases} - \frac {\sqrt {2} C_{1} k \sqrt {x{\left (t \right )}}}{\sqrt {\frac {C_{1} x{\left (t \right )}}{2 k^{2}} - 1}} + \frac {\sqrt {2} x^{\frac {3}{2}}{\left (t \right )}}{2 k \sqrt {\frac {C_{1} x{\left (t \right )}}{2 k^{2}} - 1}} + \frac {2 k^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {x{\left (t \right )}}}{2 k} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} x{\left (t \right )}}{k^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} k \sqrt {x{\left (t \right )}}}{\sqrt {- \frac {C_{1} x{\left (t \right )}}{2 k^{2}} + 1}} - \frac {\sqrt {2} i x^{\frac {3}{2}}{\left (t \right )}}{2 k \sqrt {- \frac {C_{1} x{\left (t \right )}}{2 k^{2}} + 1}} - \frac {2 i k^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {x{\left (t \right )}}}{2 k} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} + t, \ \begin {cases} - \frac {\sqrt {2} C_{1} k \sqrt {x{\left (t \right )}}}{\sqrt {\frac {C_{1} x{\left (t \right )}}{2 k^{2}} - 1}} + \frac {\sqrt {2} x^{\frac {3}{2}}{\left (t \right )}}{2 k \sqrt {\frac {C_{1} x{\left (t \right )}}{2 k^{2}} - 1}} + \frac {2 k^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {x{\left (t \right )}}}{2 k} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} x{\left (t \right )}}{k^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} k \sqrt {x{\left (t \right )}}}{\sqrt {- \frac {C_{1} x{\left (t \right )}}{2 k^{2}} + 1}} - \frac {\sqrt {2} i x^{\frac {3}{2}}{\left (t \right )}}{2 k \sqrt {- \frac {C_{1} x{\left (t \right )}}{2 k^{2}} + 1}} - \frac {2 i k^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {x{\left (t \right )}}}{2 k} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} - t\right ]
\]