60.2.240 problem 816

Internal problem ID [10814]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 816
Date solved : Sunday, March 30, 2025 at 07:06:07 PM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 226

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

\[ -\int _{\textit {\_b}}^{x}\frac {\left (-\textit {\_a} +y\right )^{3} \left (\textit {\_a} +y\right )^{3} \textit {\_a}}{-\textit {\_a}^{6}+3 \textit {\_a}^{4} y^{2}-3 \textit {\_a}^{2} y^{4}+y^{6}+\textit {\_a}^{2}-y^{2}-1}d \textit {\_a} +\int _{}^{y}\frac {2 \left (\left (-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1\right ) \int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a} -\textit {\_f} \right )^{2} \left (\textit {\_a} +\textit {\_f} \right )^{2} \textit {\_a} \left (2 \textit {\_a}^{2}-2 \textit {\_f}^{2}-3\right )}{\left (\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+\left (3 \textit {\_f}^{4}-1\right ) \textit {\_a}^{2}-\textit {\_f}^{6}+\textit {\_f}^{2}+1\right )^{2}}d \textit {\_a} +\frac {x^{2}}{2}-\frac {\textit {\_f}^{2}}{2}-\frac {1}{2}\right ) \textit {\_f}}{-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1}d \textit {\_f} +c_1 = 0 \]

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 298

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-K[2]^3+x^2 K[2]-K[2]}{-x^6+3 K[2]^2 x^4-3 K[2]^4 x^2+x^2+K[2]^6-K[2]^2-1}-\int _1^x\left (-\frac {2 K[1] K[2]}{K[1]^6-3 K[2]^2 K[1]^4+3 K[2]^4 K[1]^2-K[1]^2-K[2]^6+K[2]^2+1}-\frac {\left (K[1]^3-K[2]^2 K[1]-K[1]\right ) \left (-6 K[2]^5+12 K[1]^2 K[2]^3-6 K[1]^4 K[2]+2 K[2]\right )}{\left (K[1]^6-3 K[2]^2 K[1]^4+3 K[2]^4 K[1]^2-K[1]^2-K[2]^6+K[2]^2+1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (K[1]+\frac {K[1]^3-y(x)^2 K[1]-K[1]}{K[1]^6-3 y(x)^2 K[1]^4+3 y(x)^4 K[1]^2-K[1]^2-y(x)^6+y(x)^2+1}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out