55.2.14 problem 14

Internal problem ID [13240]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 14
Date solved : Wednesday, October 01, 2025 at 03:55:35 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 84

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

\[ y = \frac {-2 \left (-a \,x^{2}\right )^{b -\frac {1}{2}} c_1 a \,x^{2} {\mathrm e}^{a \,x^{2}}+\left (a \,x^{2}+b \right ) \left (c_1 \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-c_1 \Gamma \left (b +\frac {1}{2}\right )-1\right )}{x \left (c_1 \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-c_1 \Gamma \left (b +\frac {1}{2}\right )-1\right )} \]

✓ Mathematica. Time used: 0.51 (sec). Leaf size: 128

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

\begin{align*} y(x)&\to \frac {x^{2 b+1} \left (a x^2+b\right ) \Gamma \left (b+\frac {1}{2},-a x^2\right )-2 \left (-a x^2\right )^{b+\frac {1}{2}} \left (-e^{a x^2} x^{2 b+1}+c_1 \left (a x^2+b\right )\right )}{x^{2 b+2} \Gamma \left (b+\frac {1}{2},-a x^2\right )-2 c_1 x \left (-a x^2\right )^{b+\frac {1}{2}}}\\ y(x)&\to a x+\frac {b}{x} \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded