54.7.72 problem 1681 (book 6.90)

Internal problem ID [12921]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1681 (book 6.90)
Date solved : Wednesday, October 01, 2025 at 02:46:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y&=0 \end{align*}

✗ Maple

ode:=4*x^2*diff(diff(y(x),x),x)-x^4*diff(y(x),x)^2+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=4*y[x] - x^4*D[y[x],x]^2 + 4*x^2*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(-x**4*Derivative(y(x), x)**2 + 4*x**2*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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