20.15.5 problem 5

Internal problem ID [3831]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 5
Date solved : Tuesday, March 04, 2025 at 05:17:43 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 58

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

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+\frac {{\mathrm e}^{2 t} c_3}{3}-\frac {c_{1}}{3} \\ x_{2} \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{4 t}+{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+\frac {{\mathrm e}^{2 t} c_3}{3}+\frac {5 c_{1}}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 174

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

\begin{align*} \text {x1}(t)\to \frac {1}{8} \left (c_1 \left (2 e^{2 t}+3 e^{4 t}+3\right )-\left (e^{2 t}-1\right ) \left ((4 c_2-9 c_3) e^{2 t}-3 c_3\right )\right ) \\ \text {x2}(t)\to \frac {1}{8} \left (9 c_3 \left (e^{2 t}-1\right )^2+3 c_1 \left (2 e^{2 t}+e^{4 t}-3\right )-4 c_2 e^{2 t} \left (e^{2 t}-3\right )\right ) \\ \text {x3}(t)\to \frac {1}{8} \left (c_1 \left (2 e^{2 t}+3 e^{4 t}-5\right )+(4 c_2-6 c_3) e^{2 t}+(9 c_3-4 c_2) e^{4 t}+5 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 60

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

\[ \left [ x^{1}{\left (t \right )} = - \frac {3 C_{1}}{5} + C_{2} e^{2 t} + C_{3} e^{4 t}, \ x^{2}{\left (t \right )} = \frac {9 C_{1}}{5} + 3 C_{2} e^{2 t} + C_{3} e^{4 t}, \ x^{3}{\left (t \right )} = C_{1} + C_{2} e^{2 t} + C_{3} e^{4 t}\right ] \]