20.20.2 problem 2

Internal problem ID [3892]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 05:18:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=t \cot \left (t^{2}\right ) x_{1} \left (t \right )+\frac {t \cos \left (t^{2}\right ) x_{3} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{2} \left (t \right )}{t}-x_{3} \left (t \right )+2-t \sin \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=\csc \left (t^{2}\right ) x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )+1-t \cos \left (t \right ) \end{align*}

✗ Maple

ode:=[diff(x__1(t),t) = t*cot(t^2)*x__1(t)+1/2*t*cos(t^2)*x__3(t), diff(x__2(t),t) = 1/t*x__2(t)-x__3(t)+2-t*sin(t), diff(x__3(t),t) = csc(t^2)*x__1(t)+x__2(t)-x__3(t)+1-t*cos(t)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x1[t],t]==t*Cot[t^2]*x1[t]+t*Cos[t^2]*x3[t],D[x2[t],t]==1/t*x2[t]-x3[t]+(2-t*Sin[t]),D[x3[t],t]==Csc[t^2]*x1[t]+x2[t]-x3[t]+(1-t*Cos[t])}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-t*x__1(t)/tan(t**2) - t*x__3(t)*cos(t**2)/2 + Derivative(x__1(t), t),0),Eq(t*sin(t) + x__3(t) + Derivative(x__2(t), t) - 2 - x__2(t)/t,0),Eq(t*cos(t) - x__1(t)/sin(t**2) - x__2(t) + x__3(t) + Derivative(x__3(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

NotImplementedError :