54.2.404 problem 983

Internal problem ID [12278]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 983
Date solved : Wednesday, October 01, 2025 at 01:27:32 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 168

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

\[ y = \frac {\left (x^{2}-1\right ) \left (\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{{2}/{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{4}-18 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{{2}/{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{2}+9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{{2}/{3}} \ln \left (\frac {x +1}{x -1}\right )+6 \textit {\_Z} \sqrt {3}+3 \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (\frac {\left (\tan \left (\textit {\_Z} \right )+\sqrt {3}\right )^{3}}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )-2 \ln \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )-18 c_1 \right )\right )+1\right ) \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{{1}/{3}}}{2}+x \]

✓ Mathematica. Time used: 1.14 (sec). Leaf size: 93

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

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

✗ Sympy

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

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