80.5.60 problem D 12
Internal
problem
ID
[21281]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
5.
Second
order
equations.
Excercise
5.9
at
page
119
Problem
number
:
D
12
Date
solved
:
Thursday, October 02, 2025 at 07:27:38 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} x^{\prime \prime }&=2 {x^{\prime }}^{3} x \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 324
ode:=diff(diff(x(t),t),t) = 2*diff(x(t),t)^3*x(t);
dsolve(ode,x(t), singsol=all);
\begin{align*}
x &= c_1 \\
x &= \frac {\left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{2}/{3}}+4 c_1}{2 \left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{1}/{3}}} \\
x &= \frac {-i \sqrt {3}\, \left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_1 -\left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{2}/{3}}-4 c_1}{4 \left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{1}/{3}}} \\
x &= -\frac {-i \sqrt {3}\, \left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_1 +\left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{2}/{3}}+4 c_1}{4 \left (-12 c_2 -12 t +4 \sqrt {-4 c_1^{3}+9 c_2^{2}+18 c_2 t +9 t^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 6.532 (sec). Leaf size: 351
ode=D[x[t],{t,2}]==2*D[x[t],t]^3*x[t];
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {\sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2}}-\frac {\sqrt [3]{\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2}}{\sqrt [3]{2}}\\ x(t)&\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2\right ){}^{2/3}+\sqrt [3]{2} \left (-2-2 i \sqrt {3}\right ) c_1}{4 \sqrt [3]{\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2}}\\ x(t)&\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2\right ){}^{2/3}+2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) c_1}{4 \sqrt [3]{\sqrt {9 t^2+18 c_2 t+4 c_1{}^3+9 c_2{}^2}+3 t+3 c_2}}\\ x(t)&\to 0 \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(-2*x(t)*Derivative(x(t), t)**3 + Derivative(x(t), (t, 2)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
NotImplementedError : The given ODE 2**(2/3)*(Derivative(x(t), (t, 2))/x(t))**(1/3)/4 - 2**(2/3)*sqrt(3)*I*(Derivative(x(t), (t, 2))/x(t))**(1/3)/4 + Derivative(x(t), t) cannot be solved by the factorable group method