2.24.41 Problem 66

Internal problem ID [13605]
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 : 66
Date solved : Friday, December 19, 2025 at 08:19:33 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

Unknown ode type.

2.24.41.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 192

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

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

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-a \left (\frac {13607}{13605}+b \,x^{13605}\right ) y \left (x \right )=-\frac {a^{2} x \left (\frac {13606}{13605}+b \,x^{13605}\right )}{13605} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \left (\frac {13607}{13605}+b \,x^{13605}\right ) y \left (x \right )-\frac {a^{2} x \left (\frac {13606}{13605}+b \,x^{13605}\right )}{13605}}{y \left (x \right )} \end {array} \]

2.24.41.2 ✗ Mathematica

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

Not solved

2.24.41.3 ✗ Sympy

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