61.22.22 problem 22

Internal problem ID [12268]
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 : 22
Date solved : Friday, March 14, 2025 at 04:40:10 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=-\frac {4 x}{25}+\frac {A}{\sqrt {x}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 270

ode:=y(x)*diff(y(x),x)-y(x) = -4/25*x+A/x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {625 c_{1} \left (-\frac {A y^{2} \sqrt {x}}{2}+\frac {16 x^{4}}{625}-\frac {16 x^{3} y}{125}+\frac {6 x^{2} y^{2}}{25}-\frac {x y^{3}}{5}+\frac {y^{4}}{16}+A^{2} x +\frac {4 A y x^{{3}/{2}}}{5}-\frac {8 A \,x^{{5}/{2}}}{25}\right ) \sqrt {A \,x^{{3}/{2}}}\, \sqrt {\frac {A \sqrt {x}-\frac {4 \left (x -\frac {5 y}{4}\right )^{2}}{25}}{\sqrt {x}\, A}}+\frac {625 \left (y^{4}-10 A^{2} y\right ) x^{{3}/{2}}}{2}+500 \left (-y^{3}+4 A^{2}\right ) x^{{5}/{2}}+\frac {128 x^{{11}/{2}}}{5}+400 y^{2} x^{{7}/{2}}-160 y x^{{9}/{2}}+\frac {625 \left (-\frac {y^{5}}{2}+5 A^{2} y^{2}\right ) \sqrt {x}}{4}-400 A \,x^{4}+1500 A y x^{3}-1875 A y^{2} x^{2}+\frac {3125 A y^{3} x}{4}-3125 A^{3} x}{\sqrt {A \,x^{{3}/{2}}}\, \sqrt {\frac {A \sqrt {x}-\frac {4 \left (x -\frac {5 y}{4}\right )^{2}}{25}}{\sqrt {x}\, A}}\, \left (25 A \sqrt {x}-4 \left (x -\frac {5 y}{4}\right )^{2}\right )^{2}} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==-4/25*x+A*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(-A/sqrt(x) + 4*x/25 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out