76.10.7 problem 7

Internal problem ID [17458]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 7
Date solved : Monday, March 31, 2025 at 04:14:16 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: 0.108 (sec). Leaf size: 69

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 ) &= c_1 \,{\mathrm e}^{\sqrt {3}\, t}+c_2 \,{\mathrm e}^{-\sqrt {3}\, t} \\ y \left (t \right ) &= -c_1 \sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t}+c_2 \sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t}+2 c_1 \,{\mathrm e}^{\sqrt {3}\, t}+2 c_2 \,{\mathrm e}^{-\sqrt {3}\, t} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 144

ode={D[x[t],t]==2*x[t]-y[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 \frac {1}{6} e^{-\sqrt {3} t} \left (c_1 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )-\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )-c_2 \left (\left (2 \sqrt {3}-3\right ) e^{2 \sqrt {3} t}-3-2 \sqrt {3}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.168 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + y(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)
 

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