76.29.4 problem 4

Internal problem ID [17809]
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 : 4
Date solved : Monday, March 31, 2025 at 04:33:04 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-5 x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-4 x_{1} \left (t \right )+3 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.205 (sec). Leaf size: 46

ode:=[diff(x__1(t),t) = 5*x__1(t)-3*x__2(t)-2*x__3(t), diff(x__2(t),t) = 8*x__1(t)-5*x__2(t)-4*x__3(t), diff(x__3(t),t) = -4*x__1(t)+3*x__2(t)+3*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 77

ode={D[x1[t],t]==5*x1[t]-3*x2[t]-2*x3[t],D[x2[t],t]==8*x1[t]-5*x2[t]-4*x3[t],D[x3[t],t]==-4*x1[t]+3*x2[t]+3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t (4 c_1 t-3 c_2 t-2 c_3 t+c_1) \\ \text {x2}(t)\to e^t ((8 c_1-6 c_2-4 c_3) t+c_2) \\ \text {x3}(t)\to e^t ((-4 c_1+3 c_2+2 c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-5*x__1(t) + 3*x__2(t) + 2*x__3(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) + 5*x__2(t) + 4*x__3(t) + Derivative(x__2(t), t),0),Eq(4*x__1(t) - 3*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 4 C_{2} t e^{t} + \left (\frac {3 C_{1}}{4} + C_{2} + 4 C_{3}\right ) e^{t}, \ x^{2}{\left (t \right )} = 8 C_{2} t e^{t} + \left (C_{1} + 8 C_{3}\right ) e^{t}, \ x^{3}{\left (t \right )} = - 4 C_{2} t e^{t} - 4 C_{3} e^{t}\right ] \]