23.2.184 problem 189

Internal problem ID [5539]
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 : 189
Date solved : Tuesday, September 30, 2025 at 12:51:44 PM
CAS classification : [_quadrature]

\begin{align*} x^{2} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+1&=0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 111

ode:=x^2*(a^2-x^2)*diff(y(x),x)^2+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 120

ode=x^2(a^2-x^2) (D[y[x],x])^2+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x \sqrt {x^2-a^2} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1\\ y(x)&\to \frac {x \sqrt {x^2-a^2} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \end{align*}

✓ Sympy. Time used: 1.866 (sec). Leaf size: 37

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 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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