61.10.5 problem 18

Internal problem ID [12113]
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 : 18
Date solved : Sunday, March 30, 2025 at 10:53:50 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=diff(y(x),x) = y(x)^2+a*cos(b*x)^m*y(x)+a*cos(b*x)^m; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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