55.25.16 problem 16

Internal problem ID [13725]
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 : 16
Date solved : Thursday, October 02, 2025 at 04:33:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (2 A x y+B \,x^{2}+b \right ) y^{\prime }&=A y^{2}+k \left (A k +B \right ) x^{2}+c \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 5363

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 7.888 (sec). Leaf size: 127

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

\[ \text {Solve}\left [\frac {8 A^3 k (A k+B) \left (\frac {\arctan \left (\frac {A (k x+y(x))+B x}{\sqrt {b (A k+B)+A c}}\right )}{\sqrt {b (A k+B)+A c}}+\frac {\arctan \left (\frac {\sqrt {A} (k x-y(x))}{\sqrt {c-b k}}\right )}{\sqrt {A} \sqrt {c-b k}}\right )}{\left (4 A^2-2 A c+B c\right ) (2 A k+B)}=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out