61.24.52 problem 52

Internal problem ID [12386]
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 : 52
Date solved : Monday, March 31, 2025 at 05:27:14 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{{5}/{2}}}&=\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 189

ode:=y(x)*diff(y(x),x)-a*(2*x-1)/x^(5/2)*y(x) = 1/2*a^2*(x-1)*(3*x+1)/x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\frac {18 \left (x +\frac {3}{2}\right ) \sqrt {5}\, \sqrt {\frac {\left (x -1\right ) a +x^{{3}/{2}} y}{x \left (y \sqrt {x}+a \right )}}\, 7^{{5}/{6}} \left (\frac {\left (-3 x -1\right ) a -3 x^{{3}/{2}} y}{x \left (y \sqrt {x}+a \right )}\right )^{{1}/{6}}}{1225}+1458 x \left (\int _{}^{\frac {-\frac {18 x^{{3}/{2}} y}{35}+\frac {9 \left (-2 x -3\right ) a}{35}}{x \left (y \sqrt {x}+a \right )}}\frac {\textit {\_a} \left (5 \textit {\_a} -9\right )^{{1}/{6}} \sqrt {7 \textit {\_a} +9}}{\left (35 \textit {\_a} +18\right )^{{2}/{3}} \left (1225 \textit {\_a}^{3}-3159 \textit {\_a} -1458\right )}d \textit {\_a} +\frac {c_1}{1458}\right ) \left (-\frac {a}{x \left (y \sqrt {x}+a \right )}\right )^{{2}/{3}}}{x \left (-\frac {a}{x \left (y \sqrt {x}+a \right )}\right )^{{2}/{3}}} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-a*(2*x-1)*x^(-5/2)*y[x]==1/2*a^2*(x-1)*(3*x+1)*x^(-4); 
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**2*(x - 1)*(3*x + 1)/(2*x**4) - a*(2*x - 1)*y(x)/x**(5/2) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out