54.7.187 problem 1809 (book 6.218)

Internal problem ID [13036]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1809 (book 6.218)
Date solved : Wednesday, October 01, 2025 at 02:57:37 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 20.775 (sec). Leaf size: 306

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \end{align*}

✗ 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 + (a**2*y(x)**2 - 1)*(y(x)**2 - 1)*Derivative(y(x), (x, 2)) + (-2*a**2*y(x)**2 + a**2 + 1)*y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE sqrt(-(a**2*y(x)**4 - a**2*y(x)**2 - y(x)**2 + 1)*Derivative(y(x), (x, 2))/(-2*a**2*y(x)**3 + a**2*y(x) + b*sqrt(a**2*y(x)**4 - a**2*y(x)**2 - y(x)**2 + 1) + y(x))) + Derivative(y(x), x) cannot be solved by the factorable group method