61.13.6 problem 52

Internal problem ID [12226]
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 : Tuesday, January 28, 2025 at 07:42:38 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*}

✓ Solution by Maple

Time used: 0.487 (sec). Leaf size: 232

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

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

✓ Solution by Mathematica

Time used: 8.625 (sec). Leaf size: 132

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

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