60.7.190 problem 1812 (book 6.221)

Internal problem ID [11740]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1812 (book 6.221)
Date solved : Thursday, March 13, 2025 at 09:23:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sqrt {y^{2}+x^{2}}\, y^{\prime \prime }-a \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}&=0 \end{align*}

✓ Maple. Time used: 0.312 (sec). Leaf size: 85

ode:=(x^2+y(x)^2)^(1/2)*diff(diff(y(x),x),x)-a*(1+diff(y(x),x)^2)^(3/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x \\ y &= i x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\operatorname {RootOf}\left (\arctan \left (\textit {\_g} \right )+\int _{}^{\textit {\_Z}}\frac {1+\sqrt {a^{2} \left (\textit {\_f}^{2}+1\right )}}{\left (\textit {\_f}^{2} a^{2}+a^{2}-1\right ) \left (\textit {\_f}^{2}+1\right )}d \textit {\_f} +c_{1} \right )-\textit {\_g}}{\textit {\_g}^{2}+1}d \textit {\_g} +c_{2} \right ) x \\ \end{align*}

✗ Mathematica

ode=-(a*(1 + D[y[x],x]^2)^(3/2)) + Sqrt[x^2 + y[x]^2]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(Derivative(y(x), x)**2 + 1)**(3/2) + sqrt(x**2 + y(x)**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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