55.25.9 problem 9

Internal problem ID [13718]
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 : 9
Date solved : Sunday, October 12, 2025 at 04:45:25 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 131

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

\[ y = -\frac {2 \left (k -x \right ) d}{\tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}\, \tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}}{2}\right ) \textit {\_Z} a b -2 a d k \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 a d x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+a b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-b c \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 a b \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}}{2}\right ) a \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}+a -c} \]

✗ Mathematica

ode=(a*x*y[x]-a*k*y[x]+b*x-b*k)*D[y[x],x]==c*y[x]^2+d*x*y[x]+(b-d*k)*y[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") 
k = symbols("k") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(-c*y(x)**2 - d*x*y(x) - (b - d*k)*y(x) + (-a*k*y(x) + a*x*y(x) - b*k + b*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out