61.13.9 problem 55
Internal
problem
ID
[12229]
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
:
55
Date
solved
:
Tuesday, January 28, 2025 at 01:28:12 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 281
dsolve(diff(y(x),x)=y(x)^2+m*y(x)*cot(x)+b^2*sin(x)^(2*m),y(x), singsol=all)
\[
y = \frac {b \left (-\frac {\operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right ) \cos \left (x \right )^{2} \left (m +2\right )}{3}+\sin \left (x \right )^{2} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) \csc \left (x \right )^{6} \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \left (-c_{1} \sin \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \csc \left (x \right )^{2} b \right )+\cos \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \csc \left (x \right )^{2} b \right )\right )}{c_{1} \cos \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \csc \left (x \right )^{2} b \right )+\sin \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \csc \left (x \right )^{2} b \right )}
\]
✓ Solution by Mathematica
Time used: 3.269 (sec). Leaf size: 72
DSolve[D[y[x],x]==y[x]^2+m*y[x]*Cot[x]+b^2*Sin[x]^(2*m),y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \sqrt {b^2} \sin ^m(x) \tan \left (\frac {\sqrt {b^2} \sqrt {\cos ^2(x)} \sec (x) \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}+c_1\right )
\]