54.2.173 problem 750

Internal problem ID [12047]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 750
Date solved : Wednesday, October 01, 2025 at 12:06:11 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 57

ode:=diff(y(x),x) = 1/(6*y(x)^2+x)*(x^2+3*y(x)^2)*y(x)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {y^{2} x}{6 y^{2}+x} = \frac {\left ({\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+9\right ) x \right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) x}{54} \]

✓ Mathematica. Time used: 2.591 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{2 (x+c_1)}}{x}\right )}}{\sqrt {6}}\\ y(x)&\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{2 (x+c_1)}}{x}\right )}}{\sqrt {6}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2 + 3*y(x)**2)*y(x)/(x*(x + 6*y(x)**2)) cannot be solved by the lie group method