87.31.9 problem 9

Internal problem ID [23925]
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 321
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:46:34 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

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

\begin{align*} x(t)&\to c_1 e^{-2 t}\\ y(t)&\to e^{-2 t} (c_2-c_1 t) \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + Derivative(x(t), t),0),Eq(x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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