78.5.4 problem 2.c

Internal problem ID [21043]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 2.c
Date solved : Thursday, October 02, 2025 at 07:01:43 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.122 (sec). Leaf size: 47

ode:=[diff(x(t),t) = 2*y(t), diff(y(t),t) = -3*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 \left (t \right ) &= \frac {\sqrt {6}\, \left (\cos \left (\sqrt {6}\, t \right ) c_1 -\sin \left (\sqrt {6}\, t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 69

ode={D[x[t],t]==2*y[t],D[y[t],t]==-3*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 {2}{3}} c_2 \sin \left (\sqrt {6} t\right )\\ y(t)&\to c_2 \cos \left (\sqrt {6} t\right )-\sqrt {\frac {3}{2}} c_1 \sin \left (\sqrt {6} t\right ) \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 58

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

\[ \left [ x{\left (t \right )} = \frac {\sqrt {6} C_{1} \sin {\left (\sqrt {6} t \right )}}{3} + \frac {\sqrt {6} C_{2} \cos {\left (\sqrt {6} t \right )}}{3}, \ y{\left (t \right )} = C_{1} \cos {\left (\sqrt {6} t \right )} - C_{2} \sin {\left (\sqrt {6} t \right )}\right ] \]