71.19.4 problem 4

Internal problem ID [14525]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 4
Date solved : Monday, March 31, 2025 at 12:29:32 PM
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 )&=5 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 49

ode:=[diff(x(t),t) = -x(t)-2*y(t), diff(y(t),t) = 5*x(t)+y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (3 t \right )+c_2 \cos \left (3 t \right ) \\ y \left (t \right ) &= -\frac {3 c_1 \cos \left (3 t \right )}{2}+\frac {3 c_2 \sin \left (3 t \right )}{2}-\frac {c_1 \sin \left (3 t \right )}{2}-\frac {c_2 \cos \left (3 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 54

ode={D[x[t],t]==-x[t]-2*y[t],D[y[t],t]==5*x[t]+1*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 \cos (3 t)-\frac {1}{3} (c_1+2 c_2) \sin (3 t) \\ y(t)\to c_2 \cos (3 t)+\frac {1}{3} (5 c_1+c_2) \sin (3 t) \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 46

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(-5*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{5} + \frac {3 C_{2}}{5}\right ) \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{5} - \frac {C_{2}}{5}\right ) \sin {\left (3 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )}\right ] \]