54.1.295 problem 301

Internal problem ID [11609]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 301
Date solved : Tuesday, September 30, 2025 at 09:38:08 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (6 x y^{2}+x^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right )&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 38

ode:=(6*x*y(x)^2+x^2)*diff(y(x),x)-y(x)*(3*y(x)^2-x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {3 c_1}{2}} \sqrt {6}}{6 x \sqrt {\frac {{\mathrm e}^{3 c_1}}{x^{3} \operatorname {LambertW}\left (\frac {6 \,{\mathrm e}^{3 c_1}}{x^{3}}\right )}}} \]

✓ Mathematica. Time used: 2.203 (sec). Leaf size: 69

ode=(6*x*y[x]^2+x^2)*D[y[x],x]-y[x]*(3*y[x]^2-x)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\\ y(x)&\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 6.038 (sec). Leaf size: 24

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

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