72.10.3 problem 3

Internal problem ID [14735]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.2. page 277
Problem number : 3
Date solved : Thursday, March 13, 2025 at 04:17:56 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{-6 t} \\ y &= -{\mathrm e}^{-3 t} c_{1} +\frac {c_{2} {\mathrm e}^{-6 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

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

\begin{align*} x(t)\to \frac {1}{3} e^{-6 t} \left (c_1 \left (e^{3 t}+2\right )-2 c_2 \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-6 t} \left (c_1 \left (-e^{3 t}\right )+2 c_2 e^{3 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 32

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

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{- 6 t} - C_{2} e^{- 3 t}, \ y{\left (t \right )} = C_{1} e^{- 6 t} + C_{2} e^{- 3 t}\right ] \]