54.7.198 problem 1822 (book 6.231)

Internal problem ID [13047]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1822 (book 6.231)
Date solved : Wednesday, October 01, 2025 at 02:58:36 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x] + 3*x*D[y[x],x] + 2*y[x]*D[y[x],x]^3 + (x^2 + 2*y[x]^2*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") 
y = Function("y") 
ode = Eq(3*x*Derivative(y(x), x) + (x**2 + 2*y(x)**2*Derivative(y(x), x))*Derivative(y(x), (x, 2)) + 2*y(x)*Derivative(y(x), x)**3 + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(3*x + 2*y(x)**2*Derivative(y(x), (x, 2)))/(2*(-1/2 + sqrt(3)*I/2)*(27*(x**2*Derivative(y(x), (x, 2)) + y(x))/(4*y(x)) + sqrt(27*(3*x + 2*y(x)**2*Derivative(y(x), (x, 2)))**3/(2*y(x)**3) + 729*(x**2*Derivative(y(x), (x, 2)) + y(x))**2/(4*y(x)**2))/2)**(1/3)*y(x)) + (-1/2 + sqrt(3)*I/2)*(27*(x**2*Derivative(y(x), (x, 2)) + y(x))/(4*y(x)) + sqrt(27*(3*x + 2*y(x)**2*Derivative(y(x), (x, 2)))**3/(2*y(x)**3) + 729*(x**2*Derivative(y(x), (x, 2)) + y(x))**2/(4*y(x)**2))/2)**(1/3)/3 + Derivative(y(x), x) cannot be solved by the factorable group method