14.20.12 problem 12

Internal problem ID [3902]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:58:50 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+5 x_{2} \left (t \right )-9 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.443 (sec). Leaf size: 96

ode:=[diff(x__1(t),t) = -4*x__1(t), diff(x__2(t),t) = 2*x__1(t)+5*x__2(t)-9*x__3(t), diff(x__3(t),t) = 5*x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{-4 t} \\ x_{2} \left (t \right ) &= -\frac {c_3 \,{\mathrm e}^{-4 t}}{12}+\frac {3 \,{\mathrm e}^{2 t} \cos \left (6 t \right ) c_1}{5}+\frac {6 \,{\mathrm e}^{2 t} \cos \left (6 t \right ) c_2}{5}-\frac {6 \,{\mathrm e}^{2 t} \sin \left (6 t \right ) c_1}{5}+\frac {3 \,{\mathrm e}^{2 t} \sin \left (6 t \right ) c_2}{5} \\ x_{3} \left (t \right ) &= \frac {5 c_3 \,{\mathrm e}^{-4 t}}{36}+{\mathrm e}^{2 t} \sin \left (6 t \right ) c_2 +{\mathrm e}^{2 t} \cos \left (6 t \right ) c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 129

ode={D[x1[t],t]==-4*x1[t]-0*x2[t]+0*x3[t],D[x2[t],t]==2*x1[t]+5*x2[t]-9*x3[t],D[x3[t],t]==0*x1[t]+5*x2[t]-1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to c_1 e^{-4 t}\\ \text {x2}(t)&\to \frac {1}{12} e^{-4 t} \left ((c_1+12 c_2) e^{6 t} \cos (6 t)+3 (c_1+2 c_2-6 c_3) e^{6 t} \sin (6 t)-c_1\right )\\ \text {x3}(t)&\to \frac {1}{36} e^{-4 t} \left (-(5 c_1-36 c_3) e^{6 t} \cos (6 t)+(5 c_1+30 c_2-18 c_3) e^{6 t} \sin (6 t)+5 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 97

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) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 5*x__2(t) + 9*x__3(t) + Derivative(x__2(t), t),0),Eq(-5*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)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {36 C_{1} e^{- 4 t}}{5}, \ x^{2}{\left (t \right )} = - \frac {3 C_{1} e^{- 4 t}}{5} + \left (\frac {3 C_{2}}{5} - \frac {6 C_{3}}{5}\right ) e^{2 t} \cos {\left (6 t \right )} - \left (\frac {6 C_{2}}{5} + \frac {3 C_{3}}{5}\right ) e^{2 t} \sin {\left (6 t \right )}, \ x^{3}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{2 t} \cos {\left (6 t \right )} - C_{3} e^{2 t} \sin {\left (6 t \right )}\right ] \]