60.2.19 problem 595

Internal problem ID [10593]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 595
Date solved : Sunday, March 30, 2025 at 06:09:11 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.157 (sec). Leaf size: 92

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

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

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 204

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

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

✗ Sympy

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

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