55.13.2 problem 48

Internal problem ID [13442]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing combinations of trigonometric functions.
Problem number : 48
Date solved : Wednesday, October 01, 2025 at 12:21:00 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 256

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

\[ y = \frac {\left (-\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) \cos \left (\lambda x \right )^{\frac {n}{2}+1} \sqrt {\frac {a b}{\lambda ^{2}}}\, c_1 -\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) \lambda \sec \left (\lambda x \right )}{a \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right )} \]

✓ Mathematica. Time used: 0.467 (sec). Leaf size: 695

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

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