23.4.257 problem 257

Internal problem ID [6559]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 257
Date solved : Friday, October 03, 2025 at 02:09:31 AM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 60

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

\begin{align*} y &= -a \\ y &= a \\ y &= \frac {\left (x +\sqrt {-a^{2}+x^{2}}\right )^{c_1} c_2^{2}+\left (x +\sqrt {-a^{2}+x^{2}}\right )^{-c_1} a^{2}}{2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 60.095 (sec). Leaf size: 64

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

\begin{align*} y(x)&\to -\frac {a \tanh \left (c_2-c_1 \log \left (\sqrt {x^2-a^2}+x\right )\right )}{\sqrt {-\text {sech}^2\left (c_2-c_1 \log \left (\sqrt {x^2-a^2}+x\right )\right )}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(a - y(x))*(a + y(x))*(a**2 - x**2) + s