76.29.5 problem 5

Internal problem ID [17810]
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 : 5
Date solved : Monday, March 31, 2025 at 04:33:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-7 x_{1} \left (t \right )+9 x_{2} \left (t \right )-6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-8 x_{1} \left (t \right )+11 x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 77

ode:=[diff(x__1(t),t) = -7*x__1(t)+9*x__2(t)-6*x__3(t), diff(x__2(t),t) = -8*x__1(t)+11*x__2(t)-7*x__3(t), diff(x__3(t),t) = -2*x__1(t)+3*x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{2 t} t \\ x_{2} \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{-t}}{3}+\frac {5 c_2 \,{\mathrm e}^{2 t}}{3}+\frac {5 c_3 \,{\mathrm e}^{2 t} t}{3}-\frac {c_3 \,{\mathrm e}^{2 t}}{3} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (3 c_3 t +3 c_2 -2 c_3 \right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 158

ode={D[x1[t],t]==-7*x1[t]+9*x2[t]-6*x3[t],D[x2[t],t]==-8*x1[t]+11*x2[t]-7*x3[t],D[x3[t],t]==-2*x1[t]+3*x2[t]-1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (7-2 e^{3 t} (3 t+2)\right )+3 \left (c_2 \left (e^{3 t} (3 t+2)-2\right )-c_3 e^{3 t} (3 t+1)+c_3\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (-2 c_1 \left (e^{3 t} (15 t+7)-7\right )+3 c_2 \left (e^{3 t} (15 t+7)-4\right )-3 c_3 \left (e^{3 t} (15 t+2)-2\right )\right ) \\ \text {x3}(t)\to e^{2 t} ((-2 c_1+3 c_2-3 c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.195 (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(7*x__1(t) - 9*x__2(t) + 6*x__3(t) + Derivative(x__1(t), t),0),Eq(8*x__1(t) - 11*x__2(t) + 7*x__3(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) - 3*x__2(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {9 C_{1} t e^{2 t}}{7} + \frac {3 C_{2} e^{- t}}{2} + \left (\frac {6 C_{1}}{7} + \frac {9 C_{3}}{7}\right ) e^{2 t}, \ x^{2}{\left (t \right )} = \frac {15 C_{1} t e^{2 t}}{7} + C_{2} e^{- t} + \left (C_{1} + \frac {15 C_{3}}{7}\right ) e^{2 t}, \ x^{3}{\left (t \right )} = \frac {9 C_{1} t e^{2 t}}{7} + \frac {9 C_{3} e^{2 t}}{7}\right ] \]