54.7.76 problem 1685 (book 6.94)

Internal problem ID [12925]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1685 (book 6.94)
Date solved : Wednesday, October 01, 2025 at 02:46:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+b +x y \left (a +3 x y-2 x^{2} y^{2}\right )&=0 \end{align*}

✗ Maple

ode:=2*x^3*diff(diff(y(x),x),x)+x^2*(9+2*x*y(x))*diff(y(x),x)+b+x*y(x)*(a+3*x*y(x)-2*x^2*y(x)^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=b + x*y[x]*(a + 3*x*y[x] - 2*x^2*y[x]^2) + x^2*(9 + 2*x*y[x])*D[y[x],x] + 2*x^3*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(b + 2*x**3*Derivative(y(x), (x, 2)) + x**2*(2*x*y(x) + 9)*Derivative(y(x), x) + x*(a - 2*x**2*y(x)**2 + 3*x*y(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*x*y(x) - b + 2*x**3*y(x)**3 - 2*x**3*Derivative(y(x), (x, 2)) - 3*x**2*y(x)**2)/(x**2*(2*x*y(x) + 9)) cannot be solved by the factorable group method