20.16.4 problem 4

Internal problem ID [3837]
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 : 4
Date solved : Tuesday, March 04, 2025 at 05:17:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )-5 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 47

ode:=[diff(x__1(t),t) = x__1(t)-2*x__2(t), diff(x__2(t),t) = 5*x__1(t)-5*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 56

ode={D[x1[t],t]==x1[t]-2*x2[t],D[x2[t],t]==5*x1[t]-5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} (c_1 \cos (t)+(3 c_1-2 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-2 t} (c_2 \cos (t)+(5 c_1-3 c_2) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-5*x__1(t) + 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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