63.18.1 problem 2(a)

Internal problem ID [13128]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 2(a)
Date solved : Wednesday, March 05, 2025 at 09:17:55 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right ) \end{align*}

✓ Maple. Time used: 1.142 (sec). Leaf size: 47

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

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (\sqrt {6}\, t \right )+c_{2} \cos \left (\sqrt {6}\, t \right ) \\ y &= -\frac {\sqrt {6}\, \left (\cos \left (\sqrt {6}\, t \right ) c_{1} -\sin \left (\sqrt {6}\, t \right ) c_{2} \right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 69

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

\begin{align*} x(t)\to c_1 \cos \left (\sqrt {6} t\right )-\sqrt {\frac {3}{2}} c_2 \sin \left (\sqrt {6} t\right ) \\ y(t)\to c_2 \cos \left (\sqrt {6} t\right )+\sqrt {\frac {2}{3}} c_1 \sin \left (\sqrt {6} t\right ) \\ \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 60

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

\[ \left [ x{\left (t \right )} = - \frac {\sqrt {6} C_{1} \sin {\left (\sqrt {6} t \right )}}{2} - \frac {\sqrt {6} C_{2} \cos {\left (\sqrt {6} t \right )}}{2}, \ y{\left (t \right )} = C_{1} \cos {\left (\sqrt {6} t \right )} - C_{2} \sin {\left (\sqrt {6} t \right )}\right ] \]