60.7.81 problem 1690 (book 6.99)

Internal problem ID [11631]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1690 (book 6.99)
Date solved : Sunday, March 30, 2025 at 08:31:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{3}&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 37

ode:=x^4*diff(diff(y(x),x),x)+(-y(x)+x*diff(y(x),x))^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (-\arctan \left (\frac {1}{\sqrt {c_1 \,x^{2}-1}}\right )+c_2 \right ) x \\ y &= \left (\arctan \left (\frac {1}{\sqrt {c_1 \,x^{2}-1}}\right )+c_2 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 60.197 (sec). Leaf size: 95

ode=(-y[x] + x*D[y[x],x])^3 + x^4*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + (x*Derivative(y(x), x) - y(x))**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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