76.29.3 problem 3

Internal problem ID [17808]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 3
Date solved : Monday, March 31, 2025 at 04:33:03 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.176 (sec). Leaf size: 72

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)-x__3(t), diff(x__3(t),t) = -3*x__1(t)+2*x__2(t)+4*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 t +c_2 \right ) \\ x_{2} \left (t \right ) &= \frac {\left (c_3 \,t^{2}+2 c_2 t -2 c_3 t +2 c_1 \right ) {\mathrm e}^{2 t}}{2} \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (c_3 \,t^{2}+2 c_2 t -4 c_3 t +2 c_1 -2 c_2 -2 c_3 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 110

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[x2[t],t]==2*x1[t]+1*x2[t]-x3[t],D[x3[t],t]==-3*x1[t]+2*x2[t]+4*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} ((c_2+c_3) t-c_1 (t-1)) \\ \text {x2}(t)\to \frac {1}{2} e^{2 t} \left (c_2 \left (t^2-2 t+2\right )-(c_1 (t-4) t)+c_3 (t-2) t\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{2 t} \left ((c_1-c_2-c_3) t^2-6 c_1 t+4 (c_2+c_3) t+2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.172 (sec). Leaf size: 92

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) - x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(3*x__1(t) - 2*x__2(t) - 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{2} t e^{2 t} - \left (C_{1} - C_{2}\right ) e^{2 t}, \ x^{2}{\left (t \right )} = - \frac {C_{2} t^{2} e^{2 t}}{2} - t \left (C_{1} - 2 C_{2}\right ) e^{2 t} + \left (2 C_{1} - C_{3}\right ) e^{2 t}, \ x^{3}{\left (t \right )} = \frac {C_{2} t^{2} e^{2 t}}{2} + t \left (C_{1} - 3 C_{2}\right ) e^{2 t} - \left (3 C_{1} - C_{3}\right ) e^{2 t}\right ] \]