60.2.26 problem 602

Internal problem ID [10600]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 602
Date solved : Sunday, March 30, 2025 at 06:09:35 PM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.421 (sec). Leaf size: 167

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

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

✗ Sympy

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

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