54.2.323 problem 902

Internal problem ID [12197]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 902
Date solved : Wednesday, October 01, 2025 at 01:08:24 AM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {-x y^{2}+x^{3}-x -y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y} \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 177

ode:=diff(y(x),x) = (-x*y(x)^2+x^3-x-y(x)^6+3*y(x)^4*x^2-3*x^4*y(x)^2+x^6)/(-y(x)^2+x^2-1)/y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\left (-x +c_1 \right ) \left (4 c_1 \,x^{2}-4 x^{3}+\sqrt {4 c_1 -4 x +1}+1\right )}}{2 x -2 c_1} \\ y &= \frac {\sqrt {\left (-x +c_1 \right ) \left (4 c_1 \,x^{2}-4 x^{3}+\sqrt {4 c_1 -4 x +1}+1\right )}}{-2 x +2 c_1} \\ y &= \frac {\sqrt {\left (x -c_1 \right ) \left (-4 c_1 \,x^{2}+4 x^{3}+\sqrt {4 c_1 -4 x +1}-1\right )}}{2 x -2 c_1} \\ y &= \frac {\sqrt {\left (x -c_1 \right ) \left (-4 c_1 \,x^{2}+4 x^{3}+\sqrt {4 c_1 -4 x +1}-1\right )}}{-2 x +2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.316 (sec). Leaf size: 377

ode=D[y[x],x] == (-x + x^3 + x^6 - x*y[x]^2 - 3*x^4*y[x]^2 + 3*x^2*y[x]^4 - y[x]^6)/(y[x]*(-1 + x^2 - y[x]^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1-4 x^2}{16 x^3 (K[2]-x)^2}-\int _1^x\left (-\frac {3 \left (-4 K[2]^2-1\right )}{16 K[2]^4 (K[1]+K[2])^2}-\frac {-4 K[2]^2-1}{8 K[2]^3 (K[1]+K[2])^3}+\frac {4 K[2]^2+1}{8 (K[1]-K[2])^3 K[2]^3}-\frac {3 \left (4 K[2]^2+1\right )}{16 (K[1]-K[2])^2 K[2]^4}+\frac {1}{2 (K[1]-K[2])^2 K[2]^2}-\frac {3}{8 (K[1]-K[2])^4 K[2]^2}-\frac {1}{2 K[2]^2 (K[1]+K[2])^2}+\frac {1}{4 (K[1]-K[2])^3 K[2]^3}+\frac {1}{4 K[2]^3 (K[1]+K[2])^3}+\frac {3}{8 K[2]^2 (K[1]+K[2])^4}\right )dK[1]+\frac {4 x^2-1}{16 x^3 (x+K[2])^2}-\frac {1}{8 x^2 (K[2]-x)^3}-\frac {1}{8 x^2 (x+K[2])^3}\right )dK[2]+\int _1^x\left (\frac {-4 y(x)^2-1}{16 y(x)^3 (K[1]+y(x))^2}+\frac {4 y(x)^2+1}{16 (K[1]-y(x))^2 y(x)^3}-\frac {1}{8 (K[1]-y(x))^3 y(x)^2}-\frac {1}{8 y(x)^2 (K[1]+y(x))^3}+1\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**6 - 3*x**4*y(x)**2 + x**3 + 3*x**2*y(x)**4 - x*y(x)**2 - x - y(x)**6)/((x**2 - y(x)**2 - 1)*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out