2.24.46 Problem 72

Internal problem ID [13610]
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 1.3.3-2.
Problem number : 72
Date solved : Saturday, April 18, 2026 at 01:29:38 PM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y \left (x \right ) y^{\prime }\left (x \right )+a \left (2 b x +1\right ) {\mathrm e}^{b x} y \left (x \right )&=-a^{2} b \,x^{2} {\mathrm e}^{2 b x} \\ \end{align*}

Unknown ode type.

2.24.46.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 53

ode:=y(x)*diff(y(x),x)+a*(2*b*x+1)*exp(b*x)*y(x) = -a^2*b*x^2*exp(2*b*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (x \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right ) b -1\right ) a \,{\mathrm e}^{b x}}{b \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right )} \]

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

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

2.24.46.2 ✓ Mathematica. Time used: 0.283 (sec). Leaf size: 59

ode=y[x]*D[y[x],x]+a*(1+2*b*x)*Exp[b*x]*y[x]==-a^2*b*x^2*Exp[2*b*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [b x e^{\frac {a e^{b x}}{a b x e^{b x}+b y(x)}}=\operatorname {ExpIntegralEi}\left (\frac {a e^{b x}}{a b e^{b x} x+b y(x)}\right )+c_1,y(x)\right ] \]

2.24.46.3 ✗ Sympy

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

NotImplementedError : The given ODE -a*(-a*b*x**2*exp(b*x) + (-2*b*x - 1)*y(x))*exp(b*x)/y(x) + Deri
                                                                                   
                                                                                   
 

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')