2.22.6 Problem 7

Internal problem ID [13501]
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 : 7
Date solved : Friday, December 19, 2025 at 05:08:38 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.22.6.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 68

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

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

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

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

2.22.6.2 ✓ Mathematica. Time used: 0.247 (sec). Leaf size: 63

ode=y[x]*D[y[x],x]-y[x]==A*x^(-1)-A^2*x^(-3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [x^2 \left (-\frac {1}{A}+\frac {2 x^2 \log \left (\frac {x^2}{A+x y(x)}\right )+2 A-c_1 x^2+2 x y(x)}{\left (A-x^2+x y(x)\right )^2}\right )=0,y(x)\right ] \]

2.22.6.3 ✗ Sympy

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

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