61.13.5 problem 51

Internal problem ID [12146]
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 : 51
Date solved : Wednesday, March 05, 2025 at 04:53:38 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 103

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

\[ y = \frac {-c_{1} {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x}+\sec \left (\lambda x \right ) \lambda \left (\int {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x} \sin \left (\lambda x \right )d x \right ) c_{1} -\sec \left (\lambda x \right )}{\lambda \left (\int {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x} \sin \left (\lambda x \right )d x \right ) c_{1} -1} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out