22.4.4 problem 7.4

Internal problem ID [4536]
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 : 7.4
Date solved : Sunday, October 12, 2025 at 01:17:50 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.111 (sec). Leaf size: 37

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-2 t} \\ y \left (t \right ) &= 2 c_1 \,{\mathrm e}^{t}+\frac {c_3 \,{\mathrm e}^{-2 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 106

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

\begin{align*} x(t)&\to \frac {1}{6} e^{-2 t} \left (2 c_3 \left (e^t+2\right ) \left (e^t-1\right )^2+c_1 \left (9 e^t+e^{3 t}-4\right )+c_2 \left (3 e^t+e^{3 t}-4\right )\right )\\ y(t)&\to \frac {1}{3} e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (2 e^{3 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 36

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

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