14.22.8 problem 8

Internal problem ID [2735]
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 : 8
Date solved : Sunday, March 30, 2025 at 12:16:00 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = x__1(t)-3*x__2(t), diff(x__2(t),t) = -2*x__1(t)+2*x__2(t)]; 
ic:=x__1(0) = 0x__2(0) = 5; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 37

ode={D[ x1[t],t]==1*x1[t]-3*x2[t],D[ x2[t],t]==-2*x1[t]+2*x2[t]}; 
ic={x1[0]==0,x2[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -3 e^{-t} \left (e^{5 t}-1\right ) \\ \text {x2}(t)\to e^{-t} \left (3 e^{5 t}+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 3*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)
 

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