60.2.23 problem 599

Internal problem ID [10597]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 599
Date solved : Sunday, March 30, 2025 at 06:09:25 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 69

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

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

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 95

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

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

✗ Sympy

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

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