54.1.250 problem 255

Internal problem ID [11564]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 255
Date solved : Tuesday, September 30, 2025 at 09:20:13 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x \left (x y-3\right ) y^{\prime }+x y^{2}-y&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 74

ode:=x*(x*y(x)-3)*diff(y(x),x)+x*y(x)^2-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {3 \operatorname {LambertW}\left (\frac {\left (-x^{2}\right )^{{1}/{3}} c_1}{3}\right )}{x} \\ y &= -\frac {3 \operatorname {LambertW}\left (\frac {\left (-x^{2}\right )^{{1}/{3}} c_1 \left (-1+i \sqrt {3}\right )}{6}\right )}{x} \\ y &= -\frac {3 \operatorname {LambertW}\left (-\frac {\left (-x^{2}\right )^{{1}/{3}} c_1 \left (1+i \sqrt {3}\right )}{6}\right )}{x} \\ \end{align*}

✓ Mathematica. Time used: 5.656 (sec). Leaf size: 35

ode=x*(x*y[x]-3)*D[y[x],x]+x*y[x]^2-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.507 (sec). Leaf size: 75

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

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