54.2.179 problem 756

Internal problem ID [12053]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 756
Date solved : Wednesday, October 01, 2025 at 12:07:54 AM
CAS classification : [_rational, _Abel]

\begin{align*} y^{\prime }&=\frac {2 x^{3} y+x^{6}+x^{2} y^{2}+y^{3}}{x^{4}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 37

ode:=diff(y(x),x) = (2*x^3*y(x)+x^6+x^2*y(x)^2+y(x)^3)/x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-3+29 \operatorname {RootOf}\left (-81 \int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} +x +3 c_1 \right )\right ) x^{2}}{9} \]

✓ Mathematica. Time used: 1.097 (sec). Leaf size: 73

ode=D[y[x],x] == (x^6 + 2*x^3*y[x] + x^2*y[x]^2 + y[x]^3)/x^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 y(x)}{x^4}+\frac {1}{x^2}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^6}}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {1}{9} 29^{2/3} \left (\frac {1}{x^6}\right )^{2/3} x^5+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**6 + 2*x**3*y(x) + x**2*y(x)**2 + y(x)**3)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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