2.24.13 Problem 19

Internal problem ID [13577]
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 : 19
Date solved : Friday, December 19, 2025 at 07:19:59 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x}&=-\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x} \\ \end{align*}

Unknown ode type.

2.24.13.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 364

ode:=y(x)*diff(y(x),x)+1/2*a*(6*x-1)/x*y(x) = -1/2*a^2*(x-1)*(4*x-1)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {\sqrt {2}\, \left (\frac {i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}}{x a}\right )^{{3}/{2}} \left (-\frac {i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {7}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}}{2 x a}\right )}{8 x a}+\frac {5 \left (4 i \sqrt {2}\, x -i \sqrt {2}-6 i \sqrt {-x}+4 x +2\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {5}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}}{2 x a}\right )}{4 \left (4 i \sqrt {2}\, x +i \sqrt {2}-4 \sqrt {2}\, \sqrt {-x}-2 i \sqrt {-x}+4 x -2\right )}\right )}{2 \left (\frac {3}{2}-\frac {4 i \sqrt {2}\, x -i \sqrt {2}-6 i \sqrt {-x}+4 x +2}{2 \left (4 i \sqrt {2}\, x +i \sqrt {2}-4 \sqrt {2}\, \sqrt {-x}-2 i \sqrt {-x}+4 x -2\right )}\right ) \operatorname {hypergeom}\left (\left [-2, -1\right ], \left [-\frac {1}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}}{2 x a}\right )+\frac {4 i \left (i \sqrt {-x}\, a +2 x a +y-a \right ) \sqrt {-x}}{x a}} = 0 \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
Looking for potential symmetries 
Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\frac {a \left (6 x -1\right ) y \left (x \right )}{2 x}=-\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-\frac {a \left (6 x -1\right ) y \left (x \right )}{2 x}-\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x}}{y \left (x \right )} \end {array} \]

2.24.13.2 ✗ Mathematica

ode=y[x]*D[y[x],x]+1/2*a*(6*x-1)*1/x*y[x]==-1/2*a^2*(x-1)*(4*x-1)*1/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.24.13.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*(x - 1)*(4*x - 1)/(2*x) + a*(6*x - 1)*y(x)/(2*x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out