20.16.11 problem 11

Internal problem ID [3844]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 05:17:56 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 102

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

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

✓ Sympy. Time used: 0.131 (sec). Leaf size: 46

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

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