67.27.8 problem 38.10 (b)

Internal problem ID [17051]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (b)
Date solved : Thursday, October 02, 2025 at 01:42:21 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 68

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-2 t} \left (c_1 \left (e^{4 t}+1\right )+c_2 \left (e^{4 t}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{-2 t} \left (c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 31

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

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