63.19.1 problem 1(a)

Internal problem ID [13136]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 202
Problem number : 1(a)
Date solved : Wednesday, March 05, 2025 at 09:18:03 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )-3 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: 0.038 (sec). Leaf size: 94

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 144

ode={D[x[t],t]==-2*x[t]-3*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}{4} e^{t-2 \sqrt {3} t} \left (c_1 \left (\left (\sqrt {3}-2\right ) e^{4 \sqrt {3} t}-2-\sqrt {3}\right )+\sqrt {3} c_2 \left (e^{4 \sqrt {3} t}-1\right )\right ) \\ y(t)\to \frac {1}{12} e^{t-2 \sqrt {3} t} \left (3 c_2 \left (\left (2+\sqrt {3}\right ) e^{4 \sqrt {3} t}+2-\sqrt {3}\right )-\sqrt {3} c_1 \left (e^{4 \sqrt {3} t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + 3*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)
 

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