82.8.34 problem 36-34
Internal
problem
ID
[21905]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
II.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
36.
Nonlinear
differential
equations.
Page
1203
Problem
number
:
36-34
Date
solved
:
Thursday, October 02, 2025 at 08:06:46 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} x^{\prime \prime }&=x^{2}-4 x+\lambda \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 73
ode:=diff(diff(x(t),t),t) = x(t)^2-4*x(t)+lambda;
dsolve(ode,x(t), singsol=all);
\begin{align*}
-3 \int _{}^{x}\frac {1}{\sqrt {6 \textit {\_a}^{3}-36 \textit {\_a}^{2}+18 \textit {\_a} \lambda +9 c_1}}d \textit {\_a} -t -c_2 &= 0 \\
3 \int _{}^{x}\frac {1}{\sqrt {6 \textit {\_a}^{3}-36 \textit {\_a}^{2}+18 \textit {\_a} \lambda +9 c_1}}d \textit {\_a} -t -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.162 (sec). Leaf size: 384
ode=D[x[t],{t,2}]==x[t]^2-4*x[t]+\[Lambda];
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {4 \left (-x(t)+\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \text {$\#$1} \lambda +3 c_1\&,1\right ]\right ) \left (-x(t)+\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \text {$\#$1} \lambda +3 c_1\&,2\right ]\right ) \left (x(t)-\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \text {$\#$1} \lambda +3 c_1\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,3\right ]-x(t)}{\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,3\right ]-\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,2\right ]}}\right ),\frac {\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,2\right ]-\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,3\right ]}{\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,1\right ]-\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \lambda \text {$\#$1}+3 c_1\&,3\right ]}\right ){}^2}{\left (2 \lambda x(t)+\frac {2 x(t)^3}{3}-4 x(t)^2+c_1\right ) \left (\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \text {$\#$1} \lambda +3 c_1\&,1\right ]-\text {Root}\left [2 \text {$\#$1}^3-12 \text {$\#$1}^2+6 \text {$\#$1} \lambda +3 c_1\&,3\right ]\right )}=(t+c_2){}^2,x(t)\right ]
\]
✗ Sympy
from sympy import *
t = symbols("t")
lambda_ = symbols("lambda_")
x = Function("x")
ode = Eq(-lambda_ - x(t)**2 + 4*x(t) + Derivative(x(t), (t, 2)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
Timed Out