61.16.3 problem 21

Internal problem ID [12175]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-3. Equations containing arctangent.
Problem number : 21
Date solved : Wednesday, March 05, 2025 at 05:14:35 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 180

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

\[ y = \frac {x^{-k -1} \left (x^{k +1} {\mathrm e}^{\int \frac {x^{k +1} \arctan \left (x \right )^{n} x \lambda -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{\lambda \left (\int x^{k +1} \arctan \left (x \right )^{n}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{\lambda \left (\int x^{k +1} \arctan \left (x \right )^{n}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x +c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{\lambda \left (\int x^{k +1} \arctan \left (x \right )^{n}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{\lambda \left (\int x^{k +1} \arctan \left (x \right )^{n}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x +c_{1}} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out