54.2.202 problem 779

Internal problem ID [12076]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 779
Date solved : Wednesday, October 01, 2025 at 12:10:23 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Abel]

\begin{align*} y^{\prime }&=\frac {x^{3} y+x^{3}+x y^{2}+y^{3}}{\left (x -1\right ) x^{3}} \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 50

ode:=diff(y(x),x) = 1/(x-1)*(x^3*y(x)+x^3+x*y(x)^2+y(x)^3)/x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\ln \left (\frac {y+x}{x}\right )}{2}-\frac {\ln \left (\frac {x^{2}+y^{2}}{x^{2}}\right )}{4}+\frac {\arctan \left (\frac {y}{x}\right )}{2}-\ln \left (x -1\right )+\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 52

ode=D[y[x],x] == (x^3 + x^3*y[x] + x*y[x]^2 + y[x]^3)/((-1 + x)*x^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(K[1]+1) \left (K[1]^2+1\right )}dK[1]=\int _1^x\frac {1}{(K[2]-1) K[2]}dK[2]+c_1,y(x)\right ] \]

✓ Sympy. Time used: 11.130 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**3*y(x) + x**3 + x*y(x)**2 + y(x)**3)/(x**3*(x - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + \log {\left (x \right )} - \frac {\log {\left (1 + \frac {y^{2}{\left (x \right )}}{x^{2}} \right )}}{4} + \frac {\log {\left (1 + \frac {y{\left (x \right )}}{x} \right )}}{2} - \log {\left (x - 1 \right )} + \frac {\operatorname {atan}{\left (\frac {y{\left (x \right )}}{x} \right )}}{2} = 0 \]