54.2.84 problem 660

Internal problem ID [11958]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 660
Date solved : Tuesday, September 30, 2025 at 11:48:02 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y} \end{align*}

✓ Maple. Time used: 0.160 (sec). Leaf size: 29

ode:=diff(y(x),x) = -1/2*x-1/2*a+x^2*(x^2+2*a*x+a^2+4*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.34 (sec). Leaf size: 42

ode=D[y[x],x] == -1/2*a - x/2 + x^2*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{36} \left (-9 a^2-18 a x+4 x^6-24 c_1 x^3-9 x^2+36 c_1{}^2\right ) \end{align*}

✓ Sympy. Time used: 19.887 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a/2 - x**2*sqrt(a**2 + 2*a*x + x**2 + 4*y(x)) + x/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {a^{2}}{4} - \frac {a x}{2} - \frac {x^{2}}{4} + \frac {\left (C_{1} + x^{3}\right )^{2}}{9} \]