76.10.8 problem 8

Internal problem ID [17451]
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 : 8
Date solved : Thursday, March 13, 2025 at 10:08:37 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.061 (sec). Leaf size: 67

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 137

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

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

✓ Sympy. Time used: 0.198 (sec). Leaf size: 60

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

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