14.16.1 problem 1

Internal problem ID [3834]
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 : 1
Date solved : Tuesday, September 30, 2025 at 06:57:53 AM
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 )&=2 x_{1} \left (t \right )+2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 35

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= 2 c_1 \,{\mathrm e}^{3 t}-\frac {c_2 \,{\mathrm e}^{-2 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 72

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

\begin{align*} \text {x1}(t)&\to \frac {1}{5} e^{-2 t} \left (c_1 \left (e^{5 t}+4\right )+2 c_2 \left (e^{5 t}-1\right )\right )\\ \text {x2}(t)&\to \frac {1}{5} e^{-2 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (4 e^{5 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 34

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(-2*x__1(t) - 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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