54.2.184 problem 761

Internal problem ID [12058]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 761
Date solved : Wednesday, October 01, 2025 at 12:08:10 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {-4 x y-x^{3}+4 x^{2}-4 x +8}{8 y+2 x^{2}-8 x +8} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 18

ode:=diff(y(x),x) = (-4*x*y(x)-x^3+4*x^2-4*x+8)/(8*y(x)+2*x^2-8*x+8); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x^{2}}{4}+\operatorname {LambertW}\left ({\mathrm e}^{-x} c_1 \right )+x \]

✓ Mathematica. Time used: 0.628 (sec). Leaf size: 38

ode=D[y[x],x] == (8 - 4*x + 4*x^2 - x^3 - 4*x*y[x])/(8 - 8*x + 2*x^2 + 8*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to W\left (-e^{-x-1+c_1}\right )-\frac {x^2}{4}+x\\ y(x)&\to -\frac {1}{4} (x-4) x \end{align*}

✓ Sympy. Time used: 0.971 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (-x**3 + 4*x**2 - 4*x*y(x) - 4*x + 8)/(2*x**2 - 8*x + 8*y(x) + 8),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x^{2}}{4} + x + W\left (C_{1} e^{- x}\right ) \]