28.4.16 problem 7.16

Internal problem ID [4548]
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 : 7.16
Date solved : Sunday, March 30, 2025 at 03:26:00 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-5 x \left (t \right )+3 y \left (t \right )&=2 \,{\mathrm e}^{3 t}\\ -x \left (t \right )+\frac {d}{d t}y \left (t \right )-y \left (t \right )&=5 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.235 (sec). Leaf size: 58

ode:=[diff(x(t),t)-5*x(t)+3*y(t) = 2*exp(3*t), -x(t)+diff(y(t),t)-y(t) = 5*exp(-t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} c_2 +{\mathrm e}^{2 t} c_1 -{\mathrm e}^{-t}-4 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -2 \,{\mathrm e}^{3 t}+\frac {{\mathrm e}^{4 t} c_2}{3}+{\mathrm e}^{2 t} c_1 -2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 99

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

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (-8 e^{4 t}+(c_1-3 c_2) \left (-e^{3 t}\right )+3 (c_1-c_2) e^{5 t}-2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (-4 e^{4 t}+(c_1-3 c_2) \left (-e^{3 t}\right )+(c_1-c_2) e^{5 t}-4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 3*y(t) - 2*exp(3*t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t) - 5*exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{2 t} + 3 C_{2} e^{4 t} - 4 e^{3 t} - e^{- t}, \ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{4 t} - 2 e^{3 t} - 2 e^{- t}\right ] \]