2.24.17 Problem 28

Internal problem ID [13581]
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 : 28
Date solved : Friday, December 19, 2025 at 07:26:42 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{{7}/{5}}}&=-\frac {a^{2} \left (x -1\right ) \left (x -16\right )}{10 x^{{9}/{5}}} \\ \end{align*}

Unknown ode type.

2.24.17.1 ✓ Maple. Time used: 0.005 (sec). Leaf size: 669

ode:=y(x)*diff(y(x),x)+1/10*a*(7*x-12)/x^(7/5)*y(x) = -1/10*a^2*(x-1)*(x-16)/x^(9/5); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\frac {a \left (7 x -12\right ) y \left (x \right )}{10 x^{{7}/{5}}}=-\frac {a^{2} \left (x -1\right ) \left (x -16\right )}{10 x^{{9}/{5}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-\frac {a \left (7 x -12\right ) y \left (x \right )}{10 x^{{7}/{5}}}-\frac {a^{2} \left (x -1\right ) \left (x -16\right )}{10 x^{{9}/{5}}}}{y \left (x \right )} \end {array} \]

2.24.17.2 ✗ Mathematica

ode=y[x]*D[y[x],x]+1/10*a*(7*x-12)*x^(-7/5)*y[x]==-1/10*a^2*(x-1)*(x-16)*x^(-9/5); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

2.24.17.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*(x - 16)*(x - 1)/(10*x**(9/5)) + a*(7*x - 12)*y(x)/(10*x**(7/5)) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0