60.2.55 problem 631

Internal problem ID [10629]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 631
Date solved : Sunday, March 30, 2025 at 06:12:16 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 23

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

\[ c_{1} -x^{3}-\frac {1}{4}-\sqrt {x^{3}-6 y} = 0 \]

✓ Mathematica. Time used: 0.335 (sec). Leaf size: 31

ode=D[y[x],x] == (x^2*(1 + 2*Sqrt[x^3 - 6*y[x]]))/2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} \left (-x^6+(1-12 c_1) x^3-36 c_1{}^2\right ) \]

✓ Sympy. Time used: 1.071 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*(2*sqrt(x**3 - 6*y(x)) + 1)/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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