20.19.9 problem 10

Internal problem ID [3889]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 10
Date solved : Sunday, March 30, 2025 at 02:11:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{2} \left (t \right )-7 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{3} \left (t \right )-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-4 x_{2} \left (t \right )+9 x_{3} \left (t \right )-3 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 82

ode:=[diff(x__1(t),t) = x__1(t), diff(x__2(t),t) = 6*x__2(t)-7*x__3(t)+3*x__4(t), diff(x__3(t),t) = 3*x__3(t)-x__4(t), diff(x__4(t),t) = -4*x__2(t)+9*x__3(t)-3*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_4 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (2 c_3 \,t^{2}+2 c_2 t +4 c_3 t +2 c_1 +2 c_2 +c_3 \right )}{2} \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ x_{4} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 \,t^{2}+c_2 t -2 c_3 t +c_1 -c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 252

ode={D[x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==0*x1[t]+6*x2[t]-7*x3[t]+3*x4[t],D[x3[t],t]==0*x1[t]+0*x2[t]+3*x3[t]-x4[t],D[x4[t],t]==0*x1[t]-4*x2[t]+9*x3[t]-3*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to e^{2 t} \left (c_2 \left (2 t^2+4 t+1\right )-c_3 t (4 t+7)+c_4 t (2 t+3)\right ) \\ \text {x3}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(c_3-c_4) t+c_3\right ) \\ \text {x4}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(-4 c_2+9 c_3-5 c_4) t+c_4\right ) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to e^{2 t} \left (c_2 \left (2 t^2+4 t+1\right )-c_3 t (4 t+7)+c_4 t (2 t+3)\right ) \\ \text {x3}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(c_3-c_4) t+c_3\right ) \\ \text {x4}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(-4 c_2+9 c_3-5 c_4) t+c_4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 119

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(-x__1(t) + Derivative(x__1(t), t),0),Eq(-6*x__2(t) + 7*x__3(t) - 3*x__4(t) + Derivative(x__2(t), t),0),Eq(-3*x__3(t) + x__4(t) + Derivative(x__3(t), t),0),Eq(4*x__2(t) - 9*x__3(t) + 3*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)
 

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