14.22.2 problem 2

Internal problem ID [2729]
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 : 2
Date solved : Tuesday, March 04, 2025 at 02:40:27 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 72

ode={D[ x1[t],t]==-2*x1[t]+1*x2[t],D[ x2[t],t]==-4*x1[t]+3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_2 \left (e^{3 t}-1\right )-c_1 \left (e^{3 t}-4\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (c_2 \left (4 e^{3 t}-1\right )-4 c_1 \left (e^{3 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(4*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)
 

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