55.22.16 problem 16

Internal problem ID [13557]
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 : Sunday, October 12, 2025 at 03:42:02 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 57

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

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

✓ 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]
 

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

✗ 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