2.22.8 Problem 9

Internal problem ID [13503]
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 : Friday, December 19, 2025 at 05:09:43 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*}

Unknown ode type.

2.22.8.1 ✓ Maple. Time used: 0.001 (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);
 

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

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )}{y \left (x \right )} \end {array} \]

2.22.8.2 ✗ 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]
 

{}

2.22.8.3 ✗ 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
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '1st_power_series', 'lie_group')