61.12.1 problem 38

Internal problem ID [12133]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number : 38
Date solved : Sunday, March 30, 2025 at 11:07:05 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 204

ode:=diff(y(x),x) = y(x)^2+a*lambda+a*(lambda-a)*cot(lambda*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\csc \left (\lambda x \right ) \left (\cos \left (\lambda x \right ) \operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) a +\cos \left (\lambda x \right ) \operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_1 a -\lambda \left (c_1 \operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right )\right )}{\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_1 +\operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE a**2/tan(lambda_*x)**2 - a*lambda_ - a*lambda_/tan(lambda_*x)**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method