88.11.3 problem 3

Internal problem ID [24050]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 54
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:55:14 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 30

ode:=[diff(y(x),x) = z(x), diff(z(x),x) = y(x)]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{-x} \\ z \left (x \right ) &= c_1 \,{\mathrm e}^{x}-c_2 \,{\mathrm e}^{-x} \\ \end{align*}

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

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

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

✓ Sympy. Time used: 0.040 (sec). Leaf size: 24

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

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