54.2.298 problem 877

Internal problem ID [12172]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 877
Date solved : Wednesday, October 01, 2025 at 01:06:10 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 0.238 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to x^2+\frac {1}{-1+\sqrt {-2 x+c_1}}\\ y(x)&\to x^2-\frac {1}{1+\sqrt {-2 x+c_1}}\\ y(x)&\to x^2 \end{align*}

✓ Sympy. Time used: 1.343 (sec). Leaf size: 61

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)**2 - 2*x*y(x) - 2*x - y(x)**3)/(x**2 - y(x) - 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1} x^{2} + x^{3} - \frac {\sqrt {- 2 C_{1} - 2 x + 1}}{2} - \frac {1}{2}}{C_{1} + x}, \ y{\left (x \right )} = \frac {C_{1} x^{2} + x^{3} + \frac {\sqrt {- 2 C_{1} - 2 x + 1}}{2} - \frac {1}{2}}{C_{1} + x}\right ] \]