63.22.4 problem 4(d)

Internal problem ID [13155]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 4(d)
Date solved : Wednesday, March 05, 2025 at 09:18:24 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 )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-10 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 72

ode={D[x[t],t]==-5*x[t]+3*y[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}{7} e^{-11 t} \left (c_1 \left (6 e^{7 t}+1\right )+3 c_2 \left (e^{7 t}-1\right )\right ) \\ y(t)\to \frac {1}{7} e^{-11 t} \left (2 c_1 \left (e^{7 t}-1\right )+c_2 \left (e^{7 t}+6\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 34

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

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