54.2.412 problem 991

Internal problem ID [12286]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 991
Date solved : Wednesday, October 01, 2025 at 01:28:06 AM
CAS classification : [[_homogeneous, `class D`], _Riccati]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 29

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

\[ y = \frac {x \left (\sqrt {2}-2 \tanh \left (\left (\int F \left (x \right ) x d x +c_1 \right ) \sqrt {2}\right )\right ) \sqrt {2}}{2} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 44

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

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

✗ Sympy

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

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