60.2.58 problem 634

Internal problem ID [10632]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 634
Date solved : Sunday, March 30, 2025 at 06:12:33 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}} \end{align*}

✓ Maple. Time used: 0.150 (sec). Leaf size: 29

ode:=diff(y(x),x) = 1/2*(1+2*x^5*(4*x^2*y(x)+1)^(1/2))/x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {x^{5}+2 c_{1} x -2 \sqrt {4 x^{2} y+1}}{2 x} = 0 \]

✓ Mathematica. Time used: 0.351 (sec). Leaf size: 31

ode=D[y[x],x] == (1/2 + x^5*Sqrt[1 + 4*x^2*y[x]])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{16} \left (x^8-8 c_1 x^4-\frac {4}{x^2}+16 c_1{}^2\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x**5*sqrt(4*x**2*y(x) + 1) + 1)/(2*x**3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**5*sqrt(4*x**2*y(x) + 1) + 1/2)/x**3 cannot be solved by the factorable group method