63.23.2 problem 4

Internal problem ID [13163]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 244
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 09:18:32 PM
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 )+{\mathrm e}^{-t}\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-10 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 47

ode:=[diff(x(t),t) = -5*x(t)+3*y(t)+exp(-t), diff(y(t),t) = 2*x(t)-10*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 3 c_{2} {\mathrm e}^{-4 t}-\frac {{\mathrm e}^{-11 t} c_{1}}{2}+\frac {3 \,{\mathrm e}^{-t}}{10} \\ y &= c_{2} {\mathrm e}^{-4 t}+{\mathrm e}^{-11 t} c_{1} +\frac {{\mathrm e}^{-t}}{15} \\ \end{align*}

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 88

ode={D[x[t],t]==-5*x[t]+3*y[t]+Exp[-t],D[y[t],t]==2*x[t]-10*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{70} e^{-11 t} \left (21 e^{10 t}+30 (2 c_1+c_2) e^{7 t}+10 (c_1-3 c_2)\right ) \\ y(t)\to \frac {1}{105} e^{-11 t} \left (7 e^{10 t}+15 (2 c_1+c_2) e^{7 t}-30 (c_1-3 c_2)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.209 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) - 3*y(t) + Derivative(x(t), t) - exp(-t),0),Eq(-2*x(t) + 10*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 C_{1} e^{- 4 t} - \frac {C_{2} e^{- 11 t}}{2} + \frac {3 e^{- t}}{10}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{- 11 t} + \frac {e^{- t}}{15}\right ] \]