20.16.19 problem 19

Internal problem ID [3852]
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 : 19
Date solved : Tuesday, March 04, 2025 at 05:18:04 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*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -4\\ x_{2} \left (0\right ) = 4\\ x_{3} \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 46

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)]; 
ic:=x__1(0) = -4x__2(0) = 4x__3(0) = 4; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{4 t}-2 \,{\mathrm e}^{2 t}-3 \\ x_{2} \left (t \right ) &= 9+{\mathrm e}^{4 t}-6 \,{\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{4 t}-2 \,{\mathrm e}^{2 t}+5 \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 48

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={x1[0]==-4,x2[0]==4,x3[0]==4}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -2 e^{2 t}+e^{4 t}-3 \\ \text {x2}(t)\to \left (e^{2 t}-3\right )^2 \\ \text {x3}(t)\to -2 e^{2 t}+e^{4 t}+5 \\ \end{align*}

✓ Sympy. Time used: 0.115 (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)
 

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