54.2.67 problem 643

Internal problem ID [11941]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 643
Date solved : Tuesday, September 30, 2025 at 11:46:23 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 22

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

\[ c_{1} +\frac {x^{3}}{2}-\sqrt {x^{2}+3 y} = 0 \]

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {1}{12} \left (x^6-6 c_1 x^3-4 x^2+9 c_1{}^2\right ) \end{align*}

✗ Sympy

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

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