20.16.17 problem 17

Internal problem ID [3850]
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 : 17
Date solved : Tuesday, March 04, 2025 at 05:18:02 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 25

ode:=[diff(x__1(t),t) = -x__1(t)+4*x__2(t), diff(x__2(t),t) = 2*x__1(t)-3*x__2(t)]; 
ic:=x__1(0) = 3x__2(0) = 0; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 30

ode={D[x1[t],t]==-x1[t]+4*x2[t],D[x2[t],t]==2*x1[t]-3*x2[t]}; 
ic={x1[0]==3,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.089 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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