9.4.4 problem problem 4

Internal problem ID [968]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 4
Date solved : Tuesday, March 04, 2025 at 12:06:33 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{5 t}+{\mathrm e}^{-2 t} c_2 \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{5 t}-6 \,{\mathrm e}^{-2 t} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

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

\begin{align*} \text {x1}(t)\to \frac {1}{7} e^{-2 t} \left (c_1 \left (6 e^{7 t}+1\right )+c_2 \left (e^{7 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{7} e^{-2 t} \left (6 c_1 \left (e^{7 t}-1\right )+c_2 \left (e^{7 t}+6\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 32

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

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{6} + C_{2} e^{5 t}, \ x^{2}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{5 t}\right ] \]