54.2.370 problem 949

Internal problem ID [12244]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 949
Date solved : Wednesday, October 01, 2025 at 01:21:40 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {x^{2} y+x^{4}+2 x^{3}-3 x^{2}+x y+x +y^{3}+3 x^{2} y^{2}-3 x y^{2}+3 y x^{4}-6 x^{3} y+x^{6}-3 x^{5}}{x \left (y+x^{2}-x +1\right )} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 91

ode:=diff(y(x),x) = (x^2*y(x)+x^4+2*x^3-3*x^2+x*y(x)+x+y(x)^3+3*x^2*y(x)^2-3*x*y(x)^2+3*y(x)*x^4-6*x^3*y(x)+x^6-3*x^5)/x/(y(x)+x^2-x+1); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.292 (sec). Leaf size: 65

ode=D[y[x],x] == (x - 3*x^2 + 2*x^3 + x^4 - 3*x^5 + x^6 + x*y[x] + x^2*y[x] - 6*x^3*y[x] + 3*x^4*y[x] - 3*x*y[x]^2 + 3*x^2*y[x]^2 + y[x]^3)/(x*(1 - x + x^2 + y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.164 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**6 - 3*x**5 + 3*x**4*y(x) + x**4 - 6*x**3*y(x) + 2*x**3 + 3*x**2*y(x)**2 + x**2*y(x) - 3*x**2 - 3*x*y(x)**2 + x*y(x) + x + y(x)**3)/(x*(x**2 - x + y(x) + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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