80.7.20 problem B 12

Internal problem ID [21339]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : B 12
Date solved : Thursday, October 02, 2025 at 07:28:33 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 105

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = x(t)-y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{\sqrt {2}\, t}+\left (\frac {1}{2}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-\sqrt {2}\, t} \\ y \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {2}}{4}\right ) \sqrt {2}\, {\mathrm e}^{\sqrt {2}\, t}-\left (\frac {1}{2}-\frac {\sqrt {2}}{4}\right ) \sqrt {2}\, {\mathrm e}^{-\sqrt {2}\, t}-\left (\frac {1}{2}+\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{\sqrt {2}\, t}-\left (\frac {1}{2}-\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-\sqrt {2}\, t} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 80

ode={D[x[t],t]==x[t]+y[t],D[y[t],t]==x[t]-y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.129 (sec). Leaf size: 68

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 = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {\left (\sqrt {2} + 2\right ) e^{\sqrt {2} t}}{4} + \frac {\left (2 - \sqrt {2}\right ) e^{- \sqrt {2} t}}{4}, \ y{\left (t \right )} = \frac {\sqrt {2} e^{\sqrt {2} t}}{4} - \frac {\sqrt {2} e^{- \sqrt {2} t}}{4}\right ] \]