70.1.11 problem 2.2 (v)

Internal problem ID [14222]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.2 (v)
Date solved : Wednesday, March 05, 2025 at 10:40:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 63

ode:=[diff(x(t),t) = -y(t), diff(y(t),t) = 2*x(t)-4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (\sqrt {2}-2\right ) t}+c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t} \\ y \left (t \right ) &= \left (2+\sqrt {2}\right ) c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t}+\left (2-\sqrt {2}\right ) c_{1} {\mathrm e}^{\left (\sqrt {2}-2\right ) t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 143

ode={D[x[t],t]==-y[t],D[y[t],t]==2*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.184 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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