61.21.4 problem 4

Internal problem ID [12236]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.9. Some Transformations
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 06:13:05 PM
CAS classification : [_Riccati]

\begin{align*} x^{2} y^{\prime }&=x^{4} f \left (x \right ) y^{2}+1 \end{align*}

✗ Maple

ode:=x^2*diff(y(x),x) = x^4*f(x)*y(x)^2+1; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x^2*D[y[x],x]==x^4*f[x]*y[x]^2+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(-x**4*f(x)*y(x)**2 + x**2*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**4*f(x)*y(x)**2 + 1)/x**2 cannot be solved by the factorable group method