61.12.4 problem 41

Internal problem ID [12136]
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 : 41
Date solved : Wednesday, March 05, 2025 at 04:42:05 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 81

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

\[ y = \frac {-\left (\csc \left (\beta x \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}-\left (\int \left (\csc \left (\beta x \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1} \right ) b}{\int \left (\csc \left (\beta x \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1}} \]

✓ Mathematica. Time used: 6.265 (sec). Leaf size: 183

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

\[ \text {Solve}\left [\int _1^x\frac {e^{-2 b K[1]} \sin ^{\frac {a}{\beta }}(\beta K[1]) (-b+a \cot (\beta K[1])+y(x))}{a \beta (b+y(x))}dK[1]+\int _1^{y(x)}\left (-\frac {e^{-2 b x} \sin ^{\frac {a}{\beta }}(x \beta )}{a \beta (b+K[2])^2}-\int _1^x\left (\frac {e^{-2 b K[1]} \sin ^{\frac {a}{\beta }}(\beta K[1])}{a \beta (b+K[2])}-\frac {e^{-2 b K[1]} (-b+a \cot (\beta K[1])+K[2]) \sin ^{\frac {a}{\beta }}(\beta K[1])}{a \beta (b+K[2])^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

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

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