14.29.14 problem 14

Internal problem ID [2812]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 14
Date solved : Sunday, March 30, 2025 at 12:21:03 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {\left (x_{1} \left (t \right )^{2}+\sqrt {x_{1} \left (t \right )^{2}+4 x_{2} \left (t \right )^{2}}\right ) x_{1} \left (t \right )}{2} \end{align*}

✗ Maple

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = -1/2*(x__1(t)^2+(x__1(t)^2+4*x__2(t)^2)^(1/2))*x__1(t)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-1/2*(x1[t]^2+Sqrt[x1[t]^2+4*x2[t]^2])*x1[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq((sqrt(x__1(t)**2 + 4*x__2(t)**2) + x__1(t)**2)*x__1(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

Timed Out