87.19.8 problem 9

Internal problem ID [23678]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 149
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:44:11 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*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 69

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: 0.005 (sec). Leaf size: 139

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 (2+\sqrt {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 (\sqrt {2}-2\right ) e^{2 \sqrt {2} t}-2-\sqrt {2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 58

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)
 

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