55.24.7 problem 7

Internal problem ID [13636]
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 : 7
Date solved : Sunday, October 12, 2025 at 04:16:38 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 45

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

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

✗ Sympy

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

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