61.12.7 problem 44

Internal problem ID [12139]
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 : 44
Date solved : Sunday, March 30, 2025 at 11:09:43 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 41

ode:=diff(y(x),x) = a*cot(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = b \,x^{n}+c +\frac {1}{c_1 -a \int \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{k}d x} \]

✓ Mathematica. Time used: 1.47 (sec). Leaf size: 74

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

\begin{align*} y(x)\to \frac {1}{\frac {a \cot ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\mu +x \lambda )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
lambda_ = symbols("lambda_") 
mu = symbols("mu") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*(-b*x**n - c + y(x))**2/tan(lambda_*x + mu)**k - b*n*x**(n - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out