60.2.59 problem 635

Internal problem ID [10633]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 635
Date solved : Sunday, March 30, 2025 at 06:13:12 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 22

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

\[ c_{1} -\frac {3 x^{2}}{2}-\sqrt {x^{3}-6 y} = 0 \]

✓ Mathematica. Time used: 0.306 (sec). Leaf size: 33

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

\[ y(x)\to \frac {1}{24} \left (-9 x^4+4 x^3+36 c_1 x^2-36 c_1{}^2\right ) \]

✗ Sympy

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

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