60.2.52 problem 628

Internal problem ID [10626]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 628
Date solved : Sunday, March 30, 2025 at 06:12:00 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.094 (sec). Leaf size: 23

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

\[ c_{1} +\frac {x^{2}}{3}+\frac {4}{27}-\frac {4 \sqrt {x^{2}+3 y}}{9} = 0 \]

✓ Mathematica. Time used: 0.289 (sec). Leaf size: 32

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

\[ y(x)\to \frac {1}{48} \left (9 x^4-2 (8+27 c_1) x^2+81 c_1{}^2\right ) \]

✓ Sympy. Time used: 1.014 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = - \frac {x^{2}}{3} + \frac {3 \left (C_{1} + x^{2}\right )^{2}}{16} \]