28.4.1 problem 7.1

Internal problem ID [4533]
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 : 7.1
Date solved : Sunday, March 30, 2025 at 03:25:39 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+2 x \left (t \right )-y \left (t \right )&=0\\ x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 69

ode:=[diff(x(t),t)+2*x(t)-y(t) = 0, x(t)+diff(y(t),t)-2*y(t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 143

ode={D[x[t],t]+2*x[t]-y[t]==0,x[t]+D[y[t],t]-2*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (c_1 \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3+2 \sqrt {3}\right )+\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (c_2 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )-\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.168 (sec). Leaf size: 58

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

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