61.16.8 problem 26

Internal problem ID [12180]
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 : 26
Date solved : Wednesday, March 05, 2025 at 05:19:15 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.133 (sec). Leaf size: 29

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

\[ y = -\tan \left (-b \lambda \left (\int x^{k -1} \arctan \left (x \right )^{n}d x \right )+c_{1} \right ) b \,x^{k} \]

✓ Mathematica. Time used: 0.548 (sec). Leaf size: 48

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

\[ y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^x\lambda \arctan (K[1])^n K[1]^{k-1}dK[1]+c_1\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (b**2*cg*x**(2*k)*atan(x)**n + cg*y(x)**2*atan(x)**n + k*y(x))/x cannot be solved by the factorable group method