61.24.66 problem 66

Internal problem ID [12400]
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, March 14, 2025 at 04:45:52 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*}

✓ 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);
 

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

✗ 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

✗ 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