Internal problem ID [10560]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\sin \left (2 x \right )^{n +1} y^{\prime }-a y^{2} \sin \left (x \right )^{2 n}=b \cos \left (x \right )^{2 n}} \]
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 325
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)
\[ y \left (x \right ) = -\frac {\sin \left (2 x \right )^{n} \left (-\sin \left (x \right )^{-2 n +1-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \cos \left (x \right )^{\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \sqrt {n^{2}-a b 4^{-n}}\, c_{1} +\sin \left (x \right )^{-2 n +1-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \cos \left (x \right )^{\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} c_{1} n +\sin \left (x \right )^{-2 n +1+\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \cos \left (x \right )^{-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \sqrt {n^{2}-a b 4^{-n}}+\sin \left (x \right )^{-2 n +1+\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \cos \left (x \right )^{-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} n \right )}{a \left (\sin \left (x \right )^{-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \cos \left (x \right )^{\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} c_{1} +\cos \left (x \right )^{-\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}} \sin \left (x \right )^{\frac {\sqrt {n^{2}-a b 4^{-n}}}{2}}\right ) \sin \left (x \right )} \]
✓ Solution by Mathematica
Time used: 33.745 (sec). Leaf size: 132
DSolve[Sin[2*x]^(n+1)*y'[x]==a*y[x]^2*Sin[x]^(2*n)+b*Cos[x]^(2*n),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{\sqrt {\frac {a \cos ^{-2 n}(x) \sin ^{2 n}(x)}{b}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {2^{2 n+2} n^2}{a b}} K[1]+1}dK[1]=\frac {1}{2} b \sin ^{-n}(2 x) \cos ^{2 n}(x) \left (\log \left (\tan \left (\frac {x}{2}\right )\right )-\log \left (\cos (x) \sec ^2\left (\frac {x}{2}\right )\right )\right ) \sqrt {\frac {a \sin ^{2 n}(x) \cos ^{-2 n}(x)}{b}}+c_1,y(x)\right ] \]