64.1.19 problem 2.5

Internal problem ID [15596]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.5
Date solved : Thursday, October 02, 2025 at 10:20:51 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=x-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ y \left (t \right ) &= -\frac {2 c_1 \cos \left (2 t \right )}{5}+\frac {2 c_2 \sin \left (2 t \right )}{5}+\frac {c_1 \sin \left (2 t \right )}{5}+\frac {c_2 \cos \left (2 t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 48

ode={D[x[t],t]==x[t]-5*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 (2 t)+(c_1-5 c_2) \sin (t) \cos (t)\\ y(t)&\to c_2 \cos (2 t)+(c_1-c_2) \sin (t) \cos (t) \end{align*}

✓ Sympy. Time used: 0.046 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 5*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)
 

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