20.19.7 problem 8

Internal problem ID [3887]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 8
Date solved : Sunday, March 30, 2025 at 02:10:55 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.242 (sec). Leaf size: 1981

ode:=[diff(x__1(t),t) = x__2(t)+3*x__3(t), diff(x__2(t),t) = 2*x__1(t)+3*x__2(t)-2*x__3(t), diff(x__3(t),t) = 2*x__2(t)+2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 148

ode={D[x1[t],t]==0*x1[t]+x2[t]+3*x3[t],D[x2[t],t]==2*x1[t]+3*x2[t]-2*x3[t],D[x3[t],t]==1*x1[t]+1*x2[t]+2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{8} e^{-t} \left (c_1 \left (e^{4 t} (4 t+1)+7\right )+e^{4 t} (8 c_2 t-4 c_3 t+7 c_3)-7 c_3\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{4 t}-1\right )+(2 c_2-c_3) e^{4 t}+c_3\right ) \\ \text {x3}(t)\to \frac {1}{8} e^{-t} \left (c_1 \left (e^{4 t} (4 t+1)-1\right )+e^{4 t} (8 c_2 t-4 c_3 t+7 c_3)+c_3\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__2(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 3*x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

Timed Out