87.30.6 problem 6

Internal problem ID [23905]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 310
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:46:25 PM
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 )&=5 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 47

ode:=[diff(x(t),t) = x(t)-y(t), diff(y(t),t) = 5*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 ) &= -2 c_1 \cos \left (2 t \right )+2 c_2 \sin \left (2 t \right )+c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 50

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

✓ Sympy. Time used: 0.051 (sec). Leaf size: 44

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

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