55.23.3 problem 3

Internal problem ID [13620]
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 1.3.2.
Problem number : 3
Date solved : Sunday, October 12, 2025 at 04:13:42 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \frac {a^{2} x -\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {Ei}_{1}\left (-\textit {\_Z} \right ) a^{2} x +c_1 \,a^{2} x \right )}{a} \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 37

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

\[ \text {Solve}\left [\operatorname {ExpIntegralEi}(a (a x-y(x)))+c_1=\frac {e^{a (a x-y(x))}}{a^2 x},y(x)\right ] \]

✗ Sympy

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

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