2.24.7 Problem 7

Internal problem ID [13571]
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 : Friday, December 19, 2025 at 07:08:37 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*}

Unknown ode type.

2.24.7.1 ✓ Maple. Time used: 0.001 (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);
 

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

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-a \left (1-\frac {b}{x}\right ) y \left (x \right )=a^{2} b \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \left (1-\frac {b}{x}\right ) y \left (x \right )+a^{2} b}{y \left (x \right )} \end {array} \]

2.24.7.2 ✓ 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]
 

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

2.24.7.3 ✗ 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