23.2.164 problem 167

Internal problem ID [5519]
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 : 167
Date solved : Tuesday, September 30, 2025 at 12:50:23 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} \left (-x^{2}+1\right ) {y^{\prime }}^{2}&=1-y^{2} \end{align*}

✓ Maple. Time used: 52.038 (sec). Leaf size: 3071

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.162 (sec). Leaf size: 98

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

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

✓ Sympy. Time used: 16.148 (sec). Leaf size: 112

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

\[ \left [ - \begin {cases} 0 & \text {for}\: y{\left (x \right )} > -1 \wedge y{\left (x \right )} < 1 \end {cases} + \log {\left (\sqrt {y^{2}{\left (x \right )} - 1} + y{\left (x \right )} \right )} + \frac {\int \sqrt {\frac {y^{2}{\left (x \right )} - 1}{x^{2} - 1}}\, dx}{\sqrt {\left (y{\left (x \right )} - 1\right ) \left (y{\left (x \right )} + 1\right )}} = C_{1}, \ - \begin {cases} 0 & \text {for}\: y{\left (x \right )} > -1 \wedge y{\left (x \right )} < 1 \end {cases} + \log {\left (\sqrt {y^{2}{\left (x \right )} - 1} + y{\left (x \right )} \right )} - \frac {\int \sqrt {\frac {y^{2}{\left (x \right )} - 1}{x^{2} - 1}}\, dx}{\sqrt {\left (y{\left (x \right )} - 1\right ) \left (y{\left (x \right )} + 1\right )}} = C_{1}\right ] \]