72.15.11 problem 19 (viii)

Internal problem ID [14813]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number : 19 (viii)
Date solved : Thursday, March 13, 2025 at 04:19:26 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.055 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 91

ode={D[x[t],t]==-3*x[t]-3*y[t],D[y[t],t]==2*x[t]+1*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.174 (sec). Leaf size: 76

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

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