20.17.13 problem 13

Internal problem ID [3867]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 05:18:20 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.079 (sec). Leaf size: 78

ode:=[diff(x__1(t),t) = -2*x__1(t)+3*x__2(t), diff(x__2(t),t) = 3*x__1(t)-2*x__2(t), diff(x__3(t),t) = x__1(t)+x__3(t)+x__4(t), diff(x__4(t),t) = x__2(t)+x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{t}-c_4 \,{\mathrm e}^{-5 t} \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{t}+c_4 \,{\mathrm e}^{-5 t} \\ x_{3} \left (t \right ) &= \left (\left (\frac {1}{2} t^{2}+t \right ) c_3 +c_{2} t +c_{1} \right ) {\mathrm e}^{t}+\frac {7 c_4 \,{\mathrm e}^{-5 t}}{36} \\ x_{4} \left (t \right ) &= c_3 \,{\mathrm e}^{t} t +c_{2} {\mathrm e}^{t}-\frac {c_4 \,{\mathrm e}^{-5 t}}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 192

ode={D[x1[t],t]==-2*x1[t]+3*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==3*x1[t]-2*x2[t]+0*x3[t]+0*x4[t],D[x3[t],t]==1*x1[t]+0*x2[t]+1*x3[t]-1*x4[t],D[x4[t],t]==0*x1[t]+1*x2[t]+0*x3[t]+1*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}+1\right )+c_2 \left (e^{6 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+1\right )\right ) \\ \text {x3}(t)\to \frac {1}{72} e^{-5 t} \left (c_1 \left (e^{6 t} \left (-18 t^2+42 t+5\right )-5\right )+c_2 \left (e^{6 t} \left (-18 t^2+30 t-5\right )+5\right )+72 e^{6 t} (c_3-c_4 t)\right ) \\ \text {x4}(t)\to \frac {1}{12} e^{-5 t} \left (c_1 \left (e^{6 t} (6 t-1)+1\right )+c_2 \left (e^{6 t} (6 t+1)-1\right )+12 c_4 e^{6 t}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.192 (sec). Leaf size: 88

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(2*x__1(t) - 3*x__2(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - x__3(t) - x__4(t) + Derivative(x__3(t), t),0),Eq(-x__2(t) - 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)
 

\[ \left [ x^{1}{\left (t \right )} = 6 C_{1} e^{- 5 t} + C_{2} e^{t}, \ x^{2}{\left (t \right )} = - 6 C_{1} e^{- 5 t} + C_{2} e^{t}, \ x^{3}{\left (t \right )} = - \frac {7 C_{1} e^{- 5 t}}{6} + \frac {C_{2} t^{2} e^{t}}{2} + t \left (C_{2} + C_{4}\right ) e^{t} + \left (C_{3} + C_{4}\right ) e^{t}, \ x^{4}{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} t e^{t} + C_{4} e^{t}\right ] \]