66.15.4 problem 19(i)

Internal problem ID [16172]
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(i)
Date solved : Thursday, October 02, 2025 at 10:43:12 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 36

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

\begin{align*} x \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ y \left (t \right ) &= c_1 \cos \left (t \right )-c_2 \sin \left (t \right )-c_1 \sin \left (t \right )-c_2 \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 39

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

\begin{align*} x(t)&\to c_1 \cos (t)+(c_1+c_2) \sin (t)\\ y(t)&\to c_2 \cos (t)-(2 c_1+c_2) \sin (t) \end{align*}

✓ Sympy. Time used: 0.048 (sec). Leaf size: 34

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

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