23.2.169 problem 172

Internal problem ID [5524]
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 : 172
Date solved : Tuesday, September 30, 2025 at 12:50:28 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 52

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 43

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

\begin{align*} y(x)&\to -\sqrt {a^2-x^2}+c_1\\ y(x)&\to \sqrt {a^2-x^2}+c_1 \end{align*}

✓ Sympy. Time used: 0.424 (sec). Leaf size: 68

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

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