55.22.4 problem 4

Internal problem ID [13545]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 4
Date solved : Sunday, October 12, 2025 at 03:38:41 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 120

ode:=y(x)*diff(y(x),x)-y(x) = 2*A*(x^(1/2)+4*A+3*A^2/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {A^{2}}{y}}\, \left (3 A +\sqrt {x}\right )}{\sqrt {\frac {-3 A^{2}-4 A \sqrt {x}-x +y}{y}}\, A}\right ) \sqrt {-\frac {A^{2}}{y}}-\sqrt {\frac {-6 A^{2}-8 A \sqrt {x}-2 x +2 y}{y}}\, \sqrt {2}+c_1 \sqrt {-\frac {A^{2}}{y}}}{\sqrt {-\frac {A^{2}}{y}}} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==2*A*(x^(1/2)+4*A+3*A^2*x^(-1/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-2*A*(3*A**2/sqrt(x) + 4*A + sqrt(x)) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out