76.8.6 problem 6

Internal problem ID [17416]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 6
Date solved : Thursday, March 13, 2025 at 10:07:59 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 )&=-5 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.056 (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 \cos \left (3 t \right ) c_{1}}{2}-\frac {3 \sin \left (3 t \right ) c_{2}}{2}-\frac {c_{1} \sin \left (3 t \right )}{2}-\frac {c_{2} \cos \left (3 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 54

ode={D[x[t],t]==x[t]+2*y[t],D[y[t],t]==-5*x[t]-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: 0.093 (sec). Leaf size: 44

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)
 

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