72.15.7 problem 19 (iv)

Internal problem ID [14809]
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 (iv)
Date solved : Thursday, March 13, 2025 at 04:19:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+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: 0.032 (sec). Leaf size: 34

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 &= c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} +c_{2} \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 39

ode={D[x[t],t]==-1*x[t]+1*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 c_1 \cos (t)+(c_2-c_1) \sin (t) \\ y(t)\to c_2 (\sin (t)+\cos (t))-2 c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.088 (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)
 

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