60.2.5 problem 581

Internal problem ID [10579]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 581
Date solved : Sunday, March 30, 2025 at 06:08:17 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.236 (sec). Leaf size: 144

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

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

✗ Sympy

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

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