55.18.6 problem 35

Internal problem ID [13485]
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 : 35
Date solved : Wednesday, October 01, 2025 at 03:31:44 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 29

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

\[ y = -\tan \left (-b \lambda \int x^{k -1} \operatorname {arccot}\left (x \right )^{n}d x +c_1 \right ) b \,x^{k} \]

✓ Mathematica. Time used: 0.93 (sec). Leaf size: 48

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

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

✗ Sympy

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

Timed Out