55.13.3 problem 49

Internal problem ID [13443]
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 : 49
Date solved : Wednesday, October 01, 2025 at 12:28:06 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 15.428 (sec). Leaf size: 467

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

\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \tan (\lambda K[1]) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) y(x) \cos ^{n+1}(\lambda K[1])+\lambda y(x)^2 \cos (\lambda K[1])\right )}{(\cos (\lambda K[1]) y(x)-1)^2}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (-\frac {a \cos ^{n+1}(x \lambda ) \csc (x \lambda ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(x \lambda )\right ) \sqrt {\sin ^2(x \lambda )}}{(n+1) \lambda }\right )}{(\cos (x \lambda ) K[2]-1)^2}-\int _1^x\left (\frac {2 \exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) K[2] \cos ^{n+1}(\lambda K[1])+\lambda K[2]^2 \cos (\lambda K[1])\right ) \sin (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^3}-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (a \csc (\lambda K[1]) \cos ^{n+1}(\lambda K[1])+2 \lambda K[2] \cos (\lambda K[1])\right ) \tan (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)*cos(lambda_*x)**n + a*cos(lambda_*x)**(n - 1) - lambda_*y(x)**2*sin(lambda_*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x)*cos(lambda_*x)**n + a*cos(lambda_*x)**(n - 1) - lambda_*y(x)**2*sin(lambda_*x) + Derivative(y(x), x) cannot be solved by the lie group method