54.2.360 problem 939

Internal problem ID [12234]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 939
Date solved : Wednesday, October 01, 2025 at 01:16:10 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {-32 x y+16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 x^{2} y^{2}+96 x y^{2}-12 y x^{4}-48 x^{3} y-48 x^{2} y+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64} \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 79

ode:=diff(y(x),x) = (-32*x*y(x)+16*x^3+16*x^2-32*x-64*y(x)^3+48*x^2*y(x)^2+96*x*y(x)^2-12*y(x)*x^4-48*x^3*y(x)-48*x^2*y(x)+x^6+6*x^5+12*x^4)/(-64*y(x)+16*x^2+32*x-64); 
dsolve(ode,y(x), singsol=all);
 

\[ x +\frac {2 \ln \left (2\right )}{5}-\frac {4 \ln \left (4 y-x^{2}-2 x -4\right )}{5}+\frac {2 \ln \left (16 y^{2}+\left (-8 x^{2}-16 x +16\right ) y+x^{4}+4 x^{3}-8 x +8\right )}{5}-\frac {2 \arctan \left (-2 y+\frac {x^{2}}{2}+x -1\right )}{5}-c_1 = 0 \]

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 86

ode=D[y[x],x] == (-32*x + 16*x^2 + 16*x^3 + 12*x^4 + 6*x^5 + x^6 - 32*x*y[x] - 48*x^2*y[x] - 48*x^3*y[x] - 12*x^4*y[x] + 96*x*y[x]^2 + 48*x^2*y[x]^2 - 64*y[x]^3)/(-64 + 32*x + 16*x^2 - 64*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [(-74)^{2/3} x+36 c_1=36 \int _1^{-\frac {(-1)^{2/3} \left (5 x^2+10 x-20 y(x)+4\right )}{\sqrt [3]{74} \left (x^2+2 x-4 y(x)-4\right )}}\frac {1}{K[1]^3-\frac {33 \sqrt [3]{-1} K[1]}{74^{2/3}}+1}dK[1],y(x)\right ] \]

✓ Sympy. Time used: 10.184 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**6 + 6*x**5 - 12*x**4*y(x) + 12*x**4 - 48*x**3*y(x) + 16*x**3 + 48*x**2*y(x)**2 - 48*x**2*y(x) + 16*x**2 + 96*x*y(x)**2 - 32*x*y(x) - 32*x - 64*y(x)**3)/(16*x**2 + 32*x - 64*y(x) - 64),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x - \frac {4 \log {\left (- \frac {x^{2}}{4} - \frac {x}{2} + y{\left (x \right )} - 1 \right )}}{5} + \frac {2 \log {\left (- \frac {x^{2}}{4} - \frac {x}{2} + \left (\frac {x^{2}}{4} + \frac {x}{2} - y{\left (x \right )}\right )^{2} + y{\left (x \right )} + \frac {1}{2} \right )}}{5} - \frac {2 \operatorname {atan}{\left (\frac {x^{2}}{2} + x - 2 y{\left (x \right )} - 1 \right )}}{5} = 0 \]