54.1.256 problem 261

Internal problem ID [11570]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 261
Date solved : Tuesday, September 30, 2025 at 09:20:50 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (2 x^{2} y-x \right ) y^{\prime }-2 x y^{2}-y&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 18

ode:=(2*x^2*y(x)-x)*diff(y(x),x)-2*x*y(x)^2-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{2 \operatorname {LambertW}\left (-\frac {c_1}{2 x^{2}}\right ) x} \]

✓ Mathematica. Time used: 5.093 (sec). Leaf size: 37

ode=(2*x^2*y[x]-x)*D[y[x],x]-2*x*y[x]^2-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2 x W\left (\frac {e^{-1+\frac {9 c_1}{2^{2/3}}}}{x^2}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.546 (sec). Leaf size: 20

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

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