55.22.8 problem 8

Internal problem ID [13549]
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 : 8
Date solved : Sunday, October 12, 2025 at 03:39:31 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 76

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

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