14.22.7 problem 7

Internal problem ID [2734]
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 : 7
Date solved : Tuesday, March 04, 2025 at 02:40:32 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 3 \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t), diff(x__2(t),t) = 4*x__1(t)+x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-t}}{4}+\frac {7 \,{\mathrm e}^{3 t}}{4} \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t}}{2}+\frac {7 \,{\mathrm e}^{3 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

ode={D[ x1[t],t]==1*x1[t]+1*x2[t],D[ x2[t],t]==4*x1[t]+1*x2[t]}; 
ic={x1[0]==2,x2[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (7 e^{4 t}+1\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (7 e^{4 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.088 (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) - x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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