28.4.2 problem 7.2

Internal problem ID [4534]
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 : 7.2
Date solved : Tuesday, March 04, 2025 at 06:51:59 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )+x \left (t \right )-5 \frac {d}{d t}y \left (t \right )-4 y \left (t \right )&=0\\ -\frac {d}{d t}y \left (t \right )-2 x \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 34

ode:=[2*diff(x(t),t)+x(t)-5*diff(y(t),t)-4*y(t) = 0, -diff(y(t),t)-2*x(t)+y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x &= {\mathrm e}^{-\frac {7 t}{2}} c_{1} +c_{2} {\mathrm e}^{-t} \\ y &= \frac {4 \,{\mathrm e}^{-\frac {7 t}{2}} c_{1}}{9}+c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 86

ode={2*D[x[t],t]+x[t]-5*D[y[t],t]-4*y[t]==0,3*D[y[t],t]-2*x[t]-4*D[y[t],t]+y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-7 t/2} \left (c_1 \left (9-4 e^{5 t/2}\right )+9 c_2 \left (e^{5 t/2}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-7 t/2} \left (c_2 \left (9 e^{5 t/2}-4\right )-4 c_1 \left (e^{5 t/2}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 4*y(t) + 2*Derivative(x(t), t) - 5*Derivative(y(t), t),0),Eq(-2*x(t) + y(t) - Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {9 C_{1} e^{- \frac {7 t}{2}}}{4} + C_{2} e^{- t}, \ y{\left (t \right )} = C_{1} e^{- \frac {7 t}{2}} + C_{2} e^{- t}\right ] \]