54.7.197 problem 1820 (book 6.229)

Internal problem ID [13046]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1820 (book 6.229)
Date solved : Friday, October 03, 2025 at 03:58:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{3} y^{\prime } y^{\prime \prime }+y^{2} b&=0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 46

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int _{}^{\ln \left (x \right )}\operatorname {RootOf}\left (-a \int _{}^{\textit {\_Z}}\frac {\textit {\_a}}{\textit {\_a}^{3} a -a \,\textit {\_a}^{2}+b}d \textit {\_a} -\textit {\_b} +c_1 \right )d \textit {\_b} +c_2} \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

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