60.1.446 problem 458

Internal problem ID [10460]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 458
Date solved : Wednesday, March 05, 2025 at 10:54:16 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.037 (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 (2\right )-\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ y &= \frac {c_{1} \sqrt {-a^{2}}+\ln \left (2\right )+\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 120

ode=-1 + x^2*(-a^2 + x^2)*D[y[x],x]^2==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: 2.757 (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)
 

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