54.2.28 problem 604

Internal problem ID [11902]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 604
Date solved : Sunday, October 12, 2025 at 02:02:36 AM
CAS classification : [`x=_G(y,y')`]

\begin{align*} y^{\prime }&=\frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y} \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 47

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

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\frac {4 \textit {\_Z}^{2} x +1}{\textit {\_Z}^{2}}\right )\right ) \\ -c_1 -\frac {1}{y}-\frac {\int _{}^{4 x +\frac {1}{y^{2}}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a}}{4} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 143

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\frac {\left (\frac {8 K[1]}{K[2]}-\frac {2 \left (4 K[1] K[2]^2+1\right )}{K[2]^3}\right ) F''\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )}{F\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )^2}dK[1]+\frac {1}{K[2]^2}+\frac {1}{2 F\left (\frac {4 x K[2]^2+1}{K[2]^2}\right ) K[2]^3}\right )dK[2]+\int _1^x-\frac {1}{F\left (\frac {4 K[1] y(x)^2+1}{y(x)^2}\right )}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

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