60.2.10 problem Problem 10

Internal problem ID [15190]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 10
Date solved : Thursday, October 02, 2025 at 10:07:02 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{3} x^{\prime \prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.230 (sec). Leaf size: 52

ode:=x(t)^3*diff(diff(x(t),t),t)+1 = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {\sqrt {\left (1+\left (c_2 +t \right ) c_1 \right ) \left (-1+\left (c_2 +t \right ) c_1 \right ) c_1}}{c_1} \\ x &= -\frac {\sqrt {\left (1+\left (c_2 +t \right ) c_1 \right ) \left (-1+\left (c_2 +t \right ) c_1 \right ) c_1}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.725 (sec). Leaf size: 93

ode=x[t]^3*D[x[t],{t,2}]+1==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {\sqrt {c_1{}^2 t^2+2 c_2 c_1{}^2 t-1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}}\\ x(t)&\to \frac {\sqrt {c_1{}^2 t^2+2 c_2 c_1{}^2 t-1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}}\\ x(t)&\to \text {Indeterminate} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t)**3*Derivative(x(t), (t, 2)) + 1,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out