2.22.5 Problem 5

Internal problem ID [13500]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 5
Date solved : Friday, December 19, 2025 at 05:08:03 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \\ \end{align*}

Unknown ode type.

2.22.5.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 179

ode:=y(x)*diff(y(x),x)-y(x) = A*x+B/x-B^2/x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (-B \,x^{2} y-B^{2} x \right ) \int _{}^{-\frac {x^{2}}{2 y x +2 B}}\frac {{\mathrm e}^{\frac {2 \,\operatorname {arctanh}\left (\frac {4 A \textit {\_a} -1}{\sqrt {4 A +1}}\right )}{\sqrt {4 A +1}}} \left (4 A \,\textit {\_a}^{2}-2 \textit {\_a} -1\right )}{\textit {\_a}^{2}}d \textit {\_a} +2 \left (-y^{2} x^{2}+\left (x^{3}-2 B x \right ) y+A \,x^{4}+B \,x^{2}-B^{2}\right ) y \,{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 A \,x^{2}+y x +B}{\sqrt {4 A +1}\, \left (y x +B \right )}\right )}{\sqrt {4 A +1}}}+c_1 x \left (y x +B \right )}{x \left (y x +B \right )} = 0 \]

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 
Looking for potential symmetries 
Looking for potential symmetries 
found: 2 potential symmetries. Proceeding with integration step 
<- Abel successful
 

Maple step by step

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

2.22.5.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==A*x+B/x-B^2*x^(-3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.5.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
y = Function("y") 
ode = Eq(-A*x + B**2/x**3 - B/x + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -A*x/y(x) + B**2/(x**3*y(x)) - B/(x*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method