61.24.1 problem 1

Internal problem ID [12335]
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 : 1
Date solved : Monday, March 31, 2025 at 05:10:58 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=\left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 201

ode:=y(x)*diff(y(x),x) = (a*x+3*b)*y(x)+c*x^3-a*b*x^2-2*b^2*x; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {x \left (\frac {-a b \,x^{3}+c \,x^{4}+a \,x^{2} y-2 b^{2} x^{2}+4 b x y-2 y^{2}}{\left (-b x +y\right )^{2}}\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {-a b x +2 c \,x^{2}+a y}{\left (-b x +y\right ) \sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}} y}{\sqrt {\frac {x^{2}}{-b x +y}}\, \left (-b x +y\right ) b}-\int _{}^{\frac {x^{2}}{-b x +y}}\frac {\left (\textit {\_a}^{2} c +\textit {\_a} a -2\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} c +a}{\sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_1 = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]==(a*x+3*b)*y[x]+c*x^3-a*b*x^2-2*b^2*x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*b*x**2 + 2*b**2*x - c*x**3 - (a*x + 3*b)*y(x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out