20.17.8 problem 8

Internal problem ID [3862]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 05:18:14 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 57

ode={D[x1[t],t]==4*x1[t]-0*x2[t]+0*x3[t],D[x2[t],t]==1*x1[t]+4*x2[t]+0*x3[t],D[x3[t],t]==0*x1[t]+1*x2[t]+4*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.128 (sec). Leaf size: 54

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

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