23.2.9 problem 9

Internal problem ID [5364]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 12:36:14 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=1-y^{2} \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 29

ode:=diff(y(x),x)^2 = 1-y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -1 \\ y &= 1 \\ y &= -\sin \left (-x +c_1 \right ) \\ y &= \sin \left (-x +c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 41

ode=(D[y[x],x])^2==1-y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sin (x-c_1)\\ y(x)&\to \sin (x+c_1)\\ y(x)&\to -1\\ y(x)&\to 1\\ y(x)&\to \text {Interval}[\{-1,1\}] \end{align*}

✓ Sympy. Time used: 100.638 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**2 + Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sin {\left (C_{1} - x \right )}, \ y{\left (x \right )} = \sin {\left (C_{1} + x \right )}\right ] \]