20.16.14 problem 14

Internal problem ID [3847]
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 : 14
Date solved : Sunday, March 30, 2025 at 02:09:56 AM
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 )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+3 x_{3} \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: 0.137 (sec). Leaf size: 40

ode:=[diff(x__1(t),t) = 2*x__1(t)-x__2(t)+3*x__3(t), diff(x__2(t),t) = 2*x__1(t)-x__2(t)+3*x__3(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 +c_3 \,{\mathrm e}^{4 t} \\ x_{2} \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{4 t}+c_1 \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{4 t}-\frac {c_2}{3}+\frac {c_1}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 119

ode={D[x1[t],t]==2*x1[t]-x2[t]+3*x3[t],D[x2[t],t]==2*x1[t]-x2[t]+3*x3[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}{4} \left (2 c_1 \left (e^{4 t}+1\right )-(c_2-3 c_3) \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{4} \left (2 c_1 \left (e^{4 t}-1\right )-c_2 \left (e^{4 t}-5\right )+3 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{4} \left (2 c_1 \left (e^{4 t}-1\right )-c_2 e^{4 t}+3 c_3 e^{4 t}+c_2+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 37

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(-2*x__1(t) + x__2(t) - 3*x__3(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)
 

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