72.9.13 problem 28

Internal problem ID [14730]
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 3.1. page 258
Problem number : 28
Date solved : Thursday, March 13, 2025 at 04:17:51 AM
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 )&=3 x \left (t \right )-2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 3 \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 43

ode:=[diff(x(t),t) = -2*x(t)-3*y(t), diff(y(t),t) = 3*x(t)-2*y(t)]; 
ic:=x(0) = 2y(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} \left (-3 \sin \left (3 t \right )+2 \cos \left (3 t \right )\right ) \\ y &= {\mathrm e}^{-2 t} \left (2 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[x[t],t]==-2*x[t]-3*y[t],D[y[t],t]==3*x[t]-2*y[t]}; 
ic={x[0]==2,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-2 t} (2 \cos (3 t)-3 \sin (3 t)) \\ y(t)\to e^{-2 t} (2 \sin (3 t)+3 \cos (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 53

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(-3*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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