87.30.9 problem 9

Internal problem ID [23908]
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 : 9
Date solved : Thursday, October 02, 2025 at 09:46:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+5 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: 0.059 (sec). Leaf size: 47

ode:=[diff(x(t),t) = -3*x(t)+5*y(t), diff(y(t),t) = -x(t)+y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 55

ode={D[x[t],t]==-3*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 e^{-t} (c_1 \cos (t)+(5 c_2-2 c_1) \sin (t))\\ y(t)&\to e^{-t} (c_2 \cos (t)-(c_1-2 c_2) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*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)
 

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