76.29.11 problem 11

Internal problem ID [17816]
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 : 11
Date solved : Monday, March 31, 2025 at 04:33:15 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 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 )&=6 x_{1} \left (t \right )+4 x_{2} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-5 x_{1} \left (t \right )-2 x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2\\ x_{3} \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.196 (sec). Leaf size: 45

ode:=[diff(x__1(t),t) = 4*x__1(t)+x__2(t)+3*x__3(t), diff(x__2(t),t) = 6*x__1(t)+4*x__2(t)+6*x__3(t), diff(x__3(t),t) = -5*x__1(t)-2*x__2(t)-4*x__3(t)]; 
ic:=x__1(0) = 1x__2(0) = -2x__3(0) = 5; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (16 t +1\right ) \\ x_{2} \left (t \right ) &= -32 \,{\mathrm e}^{t}+30 \,{\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= 15 \,{\mathrm e}^{t}-16 \,{\mathrm e}^{t} t -10 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 46

ode={D[x1[t],t]==4*x1[t]+1*x2[t]+3*x3[t],D[x2[t],t]==6*x1[t]+4*x2[t]+6*x3[t],D[x3[t],t]==-5*x1[t]-2*x2[t]-4*x3[t]}; 
ic={x1[0]==1,x2[0]==-2,x3[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t (16 t+1) \\ \text {x2}(t)\to 2 e^t \left (15 e^t-16\right ) \\ \text {x3}(t)\to e^t \left (-16 t-10 e^t+15\right ) \\ \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(-4*x__1(t) - x__2(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) - 4*x__2(t) - 6*x__3(t) + Derivative(x__2(t), t),0),Eq(5*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)
 

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