61.24.73 problem 73

Internal problem ID [12407]
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 : 73
Date solved : Monday, March 31, 2025 at 05:31:48 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 130

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

\[ y = \frac {a \left (1+2 n^{2} x +\left (\tan \left (\frac {\operatorname {RootOf}\left (2 x \,n^{2} {\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n +n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 n \,{\mathrm e}^{\textit {\_a}}+{\mathrm e}^{\textit {\_Z} +\textit {\_a}}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}+2 x +1\right ) n \right ) {\mathrm e}^{\left (n +1\right ) x}}{2 n +2} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out