61.23.6 problem 6

Internal problem ID [12328]
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 1.3.2.
Problem number : 6
Date solved : Friday, March 14, 2025 at 04:43:45 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 187

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

\[ \frac {-\sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, \operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) c_{1} a^{2}+\operatorname {BesselK}\left (1, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) c_{1} -x^{{1}/{3}} \operatorname {BesselK}\left (0, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )}{-\operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )} = 0 \]

✗ Mathematica

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

Timed Out