55.22.1 problem 1

Internal problem ID [13542]
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 : 1
Date solved : Wednesday, October 01, 2025 at 04:39:25 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 30

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

\[ y = -A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-A -c_1 -x}{A}}}{A}\right )+1\right ) \]

✓ Mathematica. Time used: 60.014 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to -A \left (1+W\left (-\frac {e^{-\frac {A+x+c_1}{A}}}{A}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.313 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = A \left (- W\left (- \frac {e^{-1 + \frac {C_{1}}{A} - \frac {x}{A}}}{A}\right ) - 1\right ) \]