54.2.65 problem 641

Internal problem ID [11939]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 641
Date solved : Sunday, October 12, 2025 at 02:04:36 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}} \end{align*}

✓ Maple. Time used: 0.208 (sec). Leaf size: 31

ode:=diff(y(x),x) = 1/2*(1+2*(4*x^2*y(x)+1)^(1/2)*x^4)/x^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 33

ode=D[y[x],x] == (1/2 + x^4*Sqrt[1 + 4*x^2*y[x]])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x**4*sqrt(4*x**2*y(x) + 1) + 1)/(2*x**3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*sqrt(4*x**2*y(x) + 1) + Derivative(y(x), x) - 1/(2*x**3) cannot be solved by the factorable group method