28.4.14 problem 7.14

Internal problem ID [4546]
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 : 7.14
Date solved : Tuesday, March 04, 2025 at 06:52:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-2 x \left (t \right )+2 \frac {d}{d t}y \left (t \right )&=-4 \,{\mathrm e}^{2 t}\\ 2 \frac {d}{d t}x \left (t \right )-3 x \left (t \right )+3 \frac {d}{d t}y \left (t \right )-y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 43

ode:=[diff(x(t),t)-2*x(t)+2*diff(y(t),t) = -4*exp(2*t), 2*diff(x(t),t)-3*x(t)+3*diff(y(t),t)-y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x &= {\mathrm e}^{-2 t} c_{2} +{\mathrm e}^{t} c_{1} +5 \,{\mathrm e}^{2 t} \\ y &= -{\mathrm e}^{-2 t} c_{2} +\frac {{\mathrm e}^{t} c_{1}}{2}-{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 79

ode={D[x[t],t]-2*x[t]+2*D[y[t],t]==-4*Exp[2*t],2*D[x[t],t]-3*x[t]+3*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}{3} e^{-2 t} \left (15 e^{4 t}+2 (c_1+c_2) e^{3 t}+c_1-2 c_2\right ) \\ y(t)\to \frac {1}{3} e^{-2 t} \left (-3 e^{4 t}+(c_1+c_2) e^{3 t}-c_1+2 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 4*exp(2*t) + Derivative(x(t), t) + 2*Derivative(y(t), t),0),Eq(-3*x(t) - y(t) + 2*Derivative(x(t), t) + 3*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{t} - C_{2} e^{- 2 t} + 5 e^{2 t}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{- 2 t} - e^{2 t}\right ] \]