76.29.7 problem 7

Internal problem ID [17804]
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 : 7
Date solved : Thursday, March 13, 2025 at 10:54:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-8 x_{1} \left (t \right )-16 x_{2} \left (t \right )-16 x_{3} \left (t \right )-17 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-10 x_{2} \left (t \right )-8 x_{3} \left (t \right )-7 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-2 x_{3} \left (t \right )-3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=6 x_{1} \left (t \right )+14 x_{2} \left (t \right )+14 x_{3} \left (t \right )+14 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.189 (sec). Leaf size: 188

ode:=[diff(x__1(t),t) = -8*x__1(t)-16*x__2(t)-16*x__3(t)-17*x__4(t), diff(x__2(t),t) = -2*x__1(t)-10*x__2(t)-8*x__3(t)-7*x__4(t), diff(x__3(t),t) = -2*x__1(t)-2*x__3(t)-3*x__4(t), diff(x__4(t),t) = 6*x__1(t)+14*x__2(t)+14*x__3(t)+14*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {9 \,{\mathrm e}^{-2 t} c_{2}}{5}-\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{5}-\frac {c_4 \,{\mathrm e}^{-t} \cos \left (t \right )}{5}+\frac {8 c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{5}-\frac {8 c_4 \,{\mathrm e}^{-t} \sin \left (t \right )}{5}+c_{1} {\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= -\frac {8 \,{\mathrm e}^{-2 t} c_{2}}{5}+\frac {4 c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{5}-\frac {4 c_4 \,{\mathrm e}^{-t} \sin \left (t \right )}{5}-\frac {3 c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{5}-\frac {3 c_4 \,{\mathrm e}^{-t} \cos \left (t \right )}{5}+\frac {c_{1} {\mathrm e}^{-2 t}}{3} \\ x_{3} \left (t \right ) &= {\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{-t} \sin \left (t \right )+c_4 \,{\mathrm e}^{-t} \cos \left (t \right ) \\ x_{4} \left (t \right ) &= -\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{5}-\frac {7 c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{5}-\frac {c_4 \,{\mathrm e}^{-t} \cos \left (t \right )}{5}+\frac {7 c_4 \,{\mathrm e}^{-t} \sin \left (t \right )}{5}+\frac {6 \,{\mathrm e}^{-2 t} c_{2}}{5}-\frac {2 c_{1} {\mathrm e}^{-2 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 289

ode={D[x1[t],t]==-8*x1[t]-16*x2[t]-16*x3[t]-17*x4[t],D[x2[t],t]==-2*x1[t]-10*x2[t]-8*x3[t]-7*x4[t],D[x3[t],t]==-2*x1[t]+0*x2[t]-2*x3[t]-3*x4[t],D[x4[t],t]==6*x1[t]+14*x2[t]+14*x3[t]+14*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} \left (-(5 c_1+9 c_2+9 c_3+12 c_4) e^t \cos (t)-(c_1+7 c_2+7 c_3+5 c_4) e^t \sin (t)+6 c_1+9 c_2+9 c_3+12 c_4\right ) \\ \text {x2}(t)\to e^{-2 t} \left (-(3 c_1+7 c_2+7 c_3+8 c_4) e^t \cos (t)+(c_1-c_2-c_3+c_4) e^t \sin (t)+3 c_1+8 c_2+7 c_3+8 c_4\right ) \\ \text {x3}(t)\to e^{-2 t} \left ((c_1+5 c_2+5 c_3+4 c_4) e^t \cos (t)-(3 c_1+5 c_2+5 c_3+7 c_4) e^t \sin (t)-c_1-5 c_2-4 c_3-4 c_4\right ) \\ \text {x4}(t)\to e^{-2 t} \left ((4 c_1+6 c_2+6 c_3+9 c_4) e^t \cos (t)+(2 c_1+8 c_2+8 c_3+7 c_4) e^t \sin (t)-2 (2 c_1+3 c_2+3 c_3+4 c_4)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.335 (sec). Leaf size: 150

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(8*x__1(t) + 16*x__2(t) + 16*x__3(t) + 17*x__4(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + 10*x__2(t) + 8*x__3(t) + 7*x__4(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) + 2*x__3(t) + 3*x__4(t) + Derivative(x__3(t), t),0),Eq(-6*x__1(t) - 14*x__2(t) - 14*x__3(t) - 14*x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {3 C_{3} e^{- 2 t}}{2} + \left (\frac {3 C_{1}}{10} + \frac {11 C_{2}}{10}\right ) e^{- t} \sin {\left (t \right )} - \left (\frac {11 C_{1}}{10} - \frac {3 C_{2}}{10}\right ) e^{- t} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- t} \cos {\left (t \right )} + \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- t} \sin {\left (t \right )} - \left (\frac {C_{3}}{2} + C_{4}\right ) e^{- 2 t}, \ x^{3}{\left (t \right )} = C_{4} e^{- 2 t} - \left (\frac {C_{1}}{10} + \frac {7 C_{2}}{10}\right ) e^{- t} \cos {\left (t \right )} - \left (\frac {7 C_{1}}{10} - \frac {C_{2}}{10}\right ) e^{- t} \sin {\left (t \right )}, \ x^{4}{\left (t \right )} = C_{1} e^{- t} \cos {\left (t \right )} - C_{2} e^{- t} \sin {\left (t \right )} + C_{3} e^{- 2 t}\right ] \]