61.22.9 problem 9

Internal problem ID [12255]
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 : 9
Date solved : Monday, March 31, 2025 at 04:55:38 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }-y&=A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 82

ode:=y(x)*diff(y(x),x)-y(x) = A*(exp(2*x/A)-1); 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

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

{}

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A*(exp(2*x/A) - 1) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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