2.24.39 Problem 56

Internal problem ID [13603]
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 : 56
Date solved : Friday, December 19, 2025 at 08:13:31 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.24.39.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 143

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

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

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

Maple step by step

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

2.24.39.2 ✗ Mathematica

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

Not solved

2.24.39.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-a**2*(k + 1)*(x - 1)/x**2 - a*(x*(k + 1) - 1)*y(x)/x**2 + 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