2.22.13 Problem 16

Internal problem ID [13508]
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 : 16
Date solved : Friday, December 19, 2025 at 05:15:28 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.22.13.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 57

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

\[ \frac {\operatorname {erf}\left (\frac {\left (-x +y\right ) \sqrt {2}}{2 \sqrt {-A}}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 \,{\mathrm e}^{\frac {\left (-x +y\right )^{2}}{2 A}} \sqrt {-A}+c_1 x}{x} = 0 \]

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 
<- Abel successful
 

Maple step by step

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

2.22.13.2 ✓ Mathematica. Time used: 0.272 (sec). Leaf size: 64

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

\[ \text {Solve}\left [-\frac {x}{\sqrt {A}}=\frac {2 e^{\frac {(x-y(x))^2}{2 A}}}{\sqrt {2 \pi } \text {erfi}\left (\frac {y(x)-x}{\sqrt {2} \sqrt {A}}\right )+2 c_1},y(x)\right ] \]

2.22.13.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/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)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method