20.13.4 problem 4

Internal problem ID [3813]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 05:17:24 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )-6 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 2*x__1(t)+4*x__2(t), diff(x__2(t),t) = -4*x__1(t)-6*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

ode={D[x1[t],t]==2*x1[t]+4*x2[t],D[x2[t],t]==-4*x1[t]-6*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} (4 c_1 t+4 c_2 t+c_1) \\ \text {x2}(t)\to e^{-2 t} (c_2-4 (c_1+c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(4*x__1(t) + 6*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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