14.22.1 problem 1

Internal problem ID [2728]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 02:40:26 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=6 x_{1} \left (t \right )-3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 6*x__1(t)-3*x__2(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+c_2 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{4 t}}{3}+c_2 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

ode={D[ x1[t],t]==6*x1[t]-3*x2[t],D[ x2[t],t]==2*x1[t]+1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} \left (c_1 \left (3 e^t-2\right )-3 c_2 \left (e^t-1\right )\right ) \\ \text {x2}(t)\to e^{3 t} \left (2 c_1 \left (e^t-1\right )+c_2 \left (3-2 e^t\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-6*x__1(t) + 3*x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{3 t} + \frac {3 C_{2} e^{4 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{4 t}\right ] \]