55.24.6 problem 6

Internal problem ID [13635]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 6
Date solved : Sunday, October 12, 2025 at 04:16:31 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ y = a \left (\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {Ei}_{1}\left (-\textit {\_Z} \right ) x +c_1 x \right )-x \right ) \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 30

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

\[ \text {Solve}\left [\operatorname {ExpIntegralEi}\left (x+\frac {y(x)}{a}\right )+c_1=\frac {e^{\frac {y(x)}{a}+x}}{x},y(x)\right ] \]

✗ Sympy

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

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