87.29.17 problem 17

Internal problem ID [23897]
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 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 17
Date solved : Thursday, October 02, 2025 at 09:46:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 30

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = x__1(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{t}-c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 68

ode={D[x1[t],t]==x2[t],D[x2[t],t]==x1[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.042 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(-x1(t) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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