54.7.146 problem 1763 (book 6.172)

Internal problem ID [12995]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1763 (book 6.172)
Date solved : Wednesday, October 01, 2025 at 02:55:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=b*x*y[x]^3 + a*y[x]*D[y[x],x] - x*D[y[x],x]^2 + x*y[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*y(x)*Derivative(y(x), x) + b*x*y(x)**3 + x*y(x)*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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