28.4.30 problem 7.30

Internal problem ID [4562]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.30
Date solved : Sunday, March 30, 2025 at 03:26:19 AM
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*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 35

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

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

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

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

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (2 c_1 \left (e^{4 t}+1\right )-c_2 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (c_2 \left (e^{4 t}+1\right )-2 c_1 \left (e^{4 t}-1\right )\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)
 

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