61.25.2 problem 2

Internal problem ID [12416]
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 1.3.4-2.
Problem number : 2
Date solved : Wednesday, March 05, 2025 at 07:03:35 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (y+a x +b \right ) y^{\prime }&=\alpha y+\beta x +\gamma \end{align*}

✓ Maple. Time used: 0.703 (sec). Leaf size: 211

ode:=(y(x)+a*x+b)*diff(y(x),x) = alpha*y(x)+beta*x+gamma; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (a x +b \right ) \alpha -\beta x -\gamma \right ) \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \tan \left (\operatorname {RootOf}\left (-2 \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \ln \left (2\right )+\sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \ln \left (-\sec \left (\textit {\_Z} \right )^{2} \left (a^{2}-2 a \alpha +\alpha ^{2}+4 \beta \right ) \left (a \alpha x +b \alpha -\beta x -\gamma \right )^{2}\right )+2 c_{1} \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }+2 a \textit {\_Z} +2 \textit {\_Z} \alpha \right )\right )+\left (a x +b \right ) \alpha ^{2}+\left (-a^{2} x -a b -\beta x -\gamma \right ) \alpha +\left (\beta x -\gamma \right ) a +2 b \beta }{2 a \alpha -2 \beta } \]

✗ Mathematica

ode=(y[x]*a*x+b)*D[y[x],x]==\[Alpha]*y[x]+\[Beta]*x+\[Gamma]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

Timed Out