23.4.268 problem 268

Internal problem ID [6570]
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 : 268
Date solved : Tuesday, September 30, 2025 at 03:07:16 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 3.944 (sec). Leaf size: 1190

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

Too large to display

✗ Sympy

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

NotImplementedError : The given ODE sqrt(-(a**2*y(x)**4*Derivative(y(x), (x, 2)) - 2*a**2*y(x)**3 -