66.9.4 problem 4

Internal problem ID [16087]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 4
Date solved : Thursday, October 02, 2025 at 10:41:13 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 30

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t} \\ \end{align*}

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

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-3 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^{-3 t} \left (c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 27

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

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 3 t} + C_{2} e^{t}, \ y{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{t}\right ] \]