60.2.12 problem 588

Internal problem ID [10586]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 588
Date solved : Sunday, March 30, 2025 at 06:08:44 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 67

ode:=diff(y(x),x) = (x+F(-(x-y(x))*(x+y(x))))/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: 0.181 (sec). Leaf size: 109

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq((-x - F((-x + y(x))*(x + y(x))))/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