74.22.9 problem 9

Internal problem ID [16577]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 9
Date solved : Thursday, March 13, 2025 at 08:23:36 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 37

ode:=[diff(x(t),t) = -x(t)-2*y(t), diff(y(t),t) = 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 &= -\frac {\cos \left (t \right ) c_{1}}{2}+\frac {c_{2} \sin \left (t \right )}{2}-\frac {c_{1} \sin \left (t \right )}{2}-\frac {c_{2} \cos \left (t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 39

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

✓ Sympy. Time used: 0.074 (sec). Leaf size: 29

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

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