2.24.25 Problem 39

Internal problem ID [13589]
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 : 39
Date solved : Friday, December 19, 2025 at 07:44:02 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.24.25.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 92

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

\[ c_1 +\frac {{\mathrm e}^{\frac {a x +y}{2 a}} \sqrt {\frac {-a x +a -y}{a x +y}}\, y}{\sqrt {\frac {a}{a x +y}}\, \left (a x +y\right ) x}+\int _{}^{\frac {a}{a x +y}}\frac {{\mathrm e}^{\frac {1}{2 \textit {\_a}}} \sqrt {\textit {\_a} -1}}{\sqrt {\textit {\_a}}}d \textit {\_a} = 0 \]

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 
found: 2 potential symmetries. Proceeding with integration step 
<- Abel successful
 

Maple step by step

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

2.24.25.2 ✗ Mathematica

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

Not solved

2.24.25.3 ✗ Sympy

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

Timed Out