60.2.303 problem 881

Internal problem ID [10877]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 881
Date solved : Sunday, March 30, 2025 at 07:18:53 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {-18 x y-6 x^{3}-18 x +27 y^{3}+27 x^{2} y^{2}+9 y x^{4}+x^{6}}{27 y+9 x^{2}+27} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 77

ode:=diff(y(x),x) = (-18*x*y(x)-6*x^3-18*x+27*y(x)^3+27*x^2*y(x)^2+9*y(x)*x^4+x^6)/(27*y(x)+9*x^2+27); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-2 c_1 \,x^{2}+2 x^{3}+3 \sqrt {2 c_1 -2 x +1}+3}{6 c_1 -6 x} \\ y &= \frac {-2 c_1 \,x^{2}+2 x^{3}-3 \sqrt {2 c_1 -2 x +1}+3}{6 c_1 -6 x} \\ \end{align*}

✓ Mathematica. Time used: 0.374 (sec). Leaf size: 68

ode=D[y[x],x] == (-18*x - 6*x^3 + x^6 - 18*x*y[x] + 9*x^4*y[x] + 27*x^2*y[x]^2 + 27*y[x]^3)/(27 + 9*x^2 + 27*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {x^2}{3}+\frac {27}{-27+\sqrt {-1458 x+c_1}} \\ y(x)\to -\frac {x^2}{3}-\frac {27}{27+\sqrt {-1458 x+c_1}} \\ y(x)\to -\frac {x^2}{3} \\ \end{align*}

✓ Sympy. Time used: 2.322 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**6 + 9*x**4*y(x) - 6*x**3 + 27*x**2*y(x)**2 - 18*x*y(x) - 18*x + 27*y(x)**3)/(9*x**2 + 27*y(x) + 27),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {- 2 C_{1} x^{2} - 2 x^{3} - 3 \sqrt {- 2 C_{1} - 2 x + 1} - 3}{6 \left (C_{1} + x\right )}, \ y{\left (x \right )} = \frac {- 2 C_{1} x^{2} - 2 x^{3} + 3 \sqrt {- 2 C_{1} - 2 x + 1} - 3}{6 \left (C_{1} + x\right )}\right ] \]