2.24.37 Problem 53

Internal problem ID [13601]
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 : 53
Date solved : Friday, December 19, 2025 at 08:08:45 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{{7}/{5}}}&=\frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{{9}/{5}}} \\ \end{align*}

Unknown ode type.

2.24.37.1 ✓ Maple. Time used: 0.007 (sec). Leaf size: 156

ode:=y(x)*diff(y(x),x)+1/5*a*(-6+x)/x^(7/5)*y(x) = 2/5*a^2*(x-1)*(x+4)/x^(9/5); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 -\frac {80 \sqrt {\frac {-y x^{{2}/{5}}-a x +a}{x^{{2}/{5}} \left (y+x^{{3}/{5}} a \right )}}\, \sqrt {3}\, \left (a y x^{{2}/{5}}+\frac {x^{{4}/{5}} y^{2}}{8}+\frac {\left (y x^{{7}/{5}}-2 a \left (x +24\right ) \left (x -1\right )\right ) a}{24}\right ) \left (a y x^{{2}/{5}}+\frac {x^{{4}/{5}} y^{2}}{8}+\frac {\left (y x^{{7}/{5}}+\frac {a \left (x +4\right )^{2}}{2}\right ) a}{4}\right )}{9 \left (\frac {a}{x^{{2}/{5}} \left (y+x^{{3}/{5}} a \right )}\right )^{{5}/{2}} \left (y x^{{2}/{5}}+a x \right )^{2} \left (\left (x +4\right ) a +y x^{{2}/{5}}\right )^{2} x} = 0 \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\frac {a \left (x -6\right ) y \left (x \right )}{5 x^{{7}/{5}}}=\frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{{9}/{5}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-\frac {a \left (x -6\right ) y \left (x \right )}{5 x^{{7}/{5}}}+\frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{{9}/{5}}}}{y \left (x \right )} \end {array} \]

2.24.37.2 ✗ Mathematica

ode=y[x]*D[y[x],x]+1/5*a*(x-6)*x^(-7/5)*y[x]==2/5*a^2*(x-1)*(x+4)*x^(-9/5); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

2.24.37.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a**2*(x - 1)*(x + 4)/(5*x**(9/5)) + a*(x - 6)*y(x)/(5*x**(7/5)) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out