Internal problem ID [9807]
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]
Solve \begin {gather*} \boxed {\left (\sin ^{n +1}\left (2 x \right )\right ) y^{\prime }-a y^{2} \left (\sin ^{2 n}\relax (x )\right )-b \left (\cos ^{2 n}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 1160
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)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 85.19 (sec). Leaf size: 185
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]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} \sqrt {\frac {4^n n^2}{a b}}+\sqrt {a b-4^n n^2} \tan \left (\frac {\sqrt {a b-4^n n^2} \sin ^{-n}(2 x) \left (-b \cos ^{2 n}(x) \left (\log \left (\cos (x) \sec ^2\left (\frac {x}{2}\right )\right )-\log \left (\tan \left (\frac {x}{2}\right )\right )\right ) \sqrt {\frac {a \sin ^{2 n}(x) \cos ^{-2 n}(x)}{b}}+2 c_1 \sin ^n(2 x)\right )}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {a \sin ^{2 n}(x) \cos ^{-2 n}(x)}{b}}} \\ \end{align*}