20.20.14 problem 14

Internal problem ID [3904]
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 : 14
Date solved : Tuesday, March 04, 2025 at 05:19:09 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = 2*x__1(t)-4*x__2(t)+3*x__3(t), diff(x__2(t),t) = -9*x__1(t)-3*x__2(t)-9*x__3(t), diff(x__3(t),t) = 4*x__1(t)+4*x__2(t)+3*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 135

ode={D[x1[t],t]==2*x1[t]-4*x2[t]+3*x3[t],D[x2[t],t]==-9*x1[t]-3*x2[t]-9*x3[t],D[x3[t],t]==4*x1[t]+4*x2[t]+3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.145 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-2*x__1(t) + 4*x__2(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(9*x__1(t) + 3*x__2(t) + 9*x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - 4*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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