2.13.4 Problem 52

Internal problem ID [13412]
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 : 52
Date solved : Sunday, January 18, 2026 at 08:01:03 PM
CAS classification : [_Riccati]

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

Entering first order ode riccati solver

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

In canonical form the ODE is

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

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \frac {\left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} a y^{2} \sin \left (x \right )^{2 n}}{2 \sin \left (x \right ) \cos \left (x \right )}+\frac {b \cos \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right ) \cos \left (x \right )} \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=\frac {b \cos \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right ) \cos \left (x \right )}\), \(f_1(x)=0\) and \(f_2(x)=\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right ) \cos \left (x \right )}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right ) \cos \left (x \right )}} \tag {1} \end{align*}

Using the above substitution in the given ODE results (after some simplification) in a second order ODE to solve for \(u(x)\) which is

\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}

But

\begin{align*} f_2' &=\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} n}{\sin \left (x \right )^{2}}-\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right )^{2}}+\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \cos \left (x \right )^{2}}-\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} n \left (\cos \left (x \right )^{2}-\sin \left (x \right )^{2}\right )}{2 \sin \left (x \right )^{2} \cos \left (x \right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\frac {a^{2} \sin \left (x \right )^{4 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-3 n} 2^{-3 n} b \cos \left (x \right )^{2 n}}{8 \sin \left (x \right )^{3} \cos \left (x \right )^{3}} \end{align*}

Substituting the above terms back in equation (2) gives

\[ \frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} u^{\prime \prime }\left (x \right )}{2 \sin \left (x \right ) \cos \left (x \right )}-\left (\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} n}{\sin \left (x \right )^{2}}-\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right )^{2}}+\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \cos \left (x \right )^{2}}-\frac {a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n} n \left (\cos \left (x \right )^{2}-\sin \left (x \right )^{2}\right )}{2 \sin \left (x \right )^{2} \cos \left (x \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {a^{2} \sin \left (x \right )^{4 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-3 n} 2^{-3 n} b \cos \left (x \right )^{2 n} u \left (x \right )}{8 \sin \left (x \right )^{3} \cos \left (x \right )^{3}} = 0 \]

Unable to solve. Will ask Maple to solve this ode now.

The solution for \(u \left (x \right )\) is

\begin{equation} \tag{3} u \left (x \right ) = c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}+c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}+\frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )-2 \cot \left (\frac {x}{2}\right )}-\frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )}+\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}-\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )}-\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )} \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{\frac {u a \sin \left (x \right )^{2 n} \left (\frac {\sin \left (2 x \right )}{2}\right )^{-n} 2^{-n}}{2 \sin \left (x \right ) \cos \left (x \right )}} \\ y &= -\frac {2 \left (\frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}+\frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )-2 \cot \left (\frac {x}{2}\right )}-\frac {c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )}+\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}-\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )}-\frac {c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )}\right ) \sin \left (x \right )^{-2 n} \sin \left (x \right ) \cos \left (x \right ) \left (\sin \left (x \right ) \cos \left (x \right )\right )^{n} 2^{n}}{a \left (c_1 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}+c_2 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {2 \left (\frac {\tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}+\frac {\tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )-2 \cot \left (\frac {x}{2}\right )}-\frac {\tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )}+\frac {c_3 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} n \left (\frac {1}{2}+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \tan \left (\frac {x}{2}\right )}-\frac {c_3 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \sqrt {-4^{-n} a b +n^{2}}\, \left (1+\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {\cot \left (\frac {x}{2}\right )^{2}}{2}\right ) \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}}{2 \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )}-\frac {c_3 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}} n \sec \left (\frac {x}{2}\right )^{2} \tan \left (\frac {x}{2}\right )}{2 \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )}\right ) \sin \left (x \right )^{-2 n} \sin \left (x \right ) \cos \left (x \right ) \left (\sin \left (x \right ) \cos \left (x \right )\right )^{n} 2^{n}}{a \left (\tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}+c_3 \tan \left (\frac {x}{2}\right )^{\frac {n}{2}} \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}} \left (-2+\sec \left (\frac {x}{2}\right )^{2}\right )^{-\frac {n}{2}}\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {\left (-c_3 \left (n +\sqrt {-4^{-n} a b +n^{2}}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}}+\left (-n +\sqrt {-4^{-n} a b +n^{2}}\right ) \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}}\right ) \sin \left (x \right )^{-2 n} \sin \left (2 x \right )^{n}}{a \left (\left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}}+c_3 \left (\tan \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )^{-\frac {\sqrt {-4^{-n} a b +n^{2}}}{2}}\right )} \\ \end{align*}

The above solution was found not to satisfy the ode or the IC. Hence it is removed.

2.13.4.1 ✓ Maple. Time used: 0.076 (sec). Leaf size: 232

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

\[ y = -\frac {\sin \left (2 x \right )^{n} \left (\sin \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} \left (n +\sqrt {n^{2}-4^{-n} b a}\right ) \cos \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}}-\cos \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} \sin \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} c_1 \left (-n +\sqrt {n^{2}-4^{-n} b a}\right )\right ) \sin \left (x \right )^{1-2 n} \csc \left (x \right )}{a \left (\cos \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} \sin \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} c_1 +\cos \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} \sin \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}}\right )} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (sin(x)^2*n+cos(x)^2 
*n+sin(x)^2-cos(x)^2)/cos(x)/sin(x)*diff(y(x),x)-1/4/cos(x)*sin(x)^(-1+2*n)*a*( 
(2*sin(x)*cos(x))^(-n))^2*cos(x)^(-1+2*n)*b/sin(x)*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\ 
(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      -> Trying changes of variables to rationalize or make the ODE simpler 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a Liouvillian solution using Kovacics algorithm 
            A Liouvillian solution exists 
            Group is reducible or imprimitive 
         <- Kovacics algorithm successful 
         Change of variables used: 
            [x = arccos(t)] 
         Linear ODE actually solved: 
            b*a*(-t^2+1)^n*t^(2*n)*u(t)+2^(2*n+2)*t^(1+2*n)*(-t^2+1)^(n+1)*(-3*\ 
t^2+n+1)*diff(u(t),t)+2^(2*n+2)*t^(2*n+2)*(-t^2+1)^(n+2)*diff(diff(u(t),t),t) =\ 
 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

2.13.4.2 ✓ Mathematica. Time used: 6.493 (sec). Leaf size: 132

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

\[ \text {Solve}\left [\int _1^{\sqrt {\frac {a \cos ^{-2 n}(x) \sin ^{2 n}(x)}{b}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {2^{2 n+2} n^2}{a b}} K[1]+1}dK[1]=\frac {1}{2} b \sin ^{-n}(2 x) \cos ^{2 n}(x) \left (\log \left (\tan \left (\frac {x}{2}\right )\right )-\log \left (\cos (x) \sec ^2\left (\frac {x}{2}\right )\right )\right ) \sqrt {\frac {a \sin ^{2 n}(x) \cos ^{-2 n}(x)}{b}}+c_1,y(x)\right ] \]

2.13.4.3 ✗ Sympy

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

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0