54.7.151 problem 1768 (book 6.177)

Internal problem ID [13000]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1768 (book 6.177)
Date solved : Friday, October 03, 2025 at 03:58:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.159 (sec). Leaf size: 55

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

\begin{align*} y &= 0 \\ y &= c_2 \,{\mathrm e}^{-\int \frac {x \sqrt {b^{2}-x^{2}}}{c_1 \sqrt {b^{2}-x^{2}}+a \left (b^{2}-x^{2}\right )}d x} \\ \end{align*}

✓ Mathematica. Time used: 7.582 (sec). Leaf size: 54

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

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

✗ Sympy

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

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