61.10.10 problem 23

Internal problem ID [12118]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with cosine.
Problem number : 23
Date solved : Sunday, March 30, 2025 at 10:56:02 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 1.089 (sec). Leaf size: 92

ode=D[y[x],x]==a*Cos[\[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 \sqrt {\sin ^2(\mu +\lambda x)} \csc (\mu +\lambda x) \cos ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(x \lambda +\mu )\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*cos(lambda_*x + mu)**k - b*n*x**(n - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out