61.13.9 problem 55
Internal
problem
ID
[12150]
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
:
Sunday, March 30, 2025 at 11:20:13 PM
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*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 281
ode:=diff(y(x),x) = y(x)^2+m*y(x)*cot(x)+b^2*sin(x)^(2*m);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-c_1 \sin \left (b \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} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right )+\cos \left (b \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} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right )\right ) \sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (-\frac {\operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {m}{2}+2\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right ) \cos \left (x \right )^{2} \left (m +2\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \sin \left (x \right )^{2}\right ) \csc \left (x \right )^{6} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} b}{c_1 \cos \left (b \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} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right )+\sin \left (b \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} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right )}
\]
✓ Mathematica. Time used: 3.269 (sec). Leaf size: 72
ode=D[y[x],x]==y[x]^2+m*y[x]*Cot[x]+b^2*Sin[x]^(2*m);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \sqrt {b^2} \sin ^m(x) \tan \left (\frac {\sqrt {b^2} \sqrt {\cos ^2(x)} \sec (x) \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}+c_1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
b = symbols("b")
m = symbols("m")
y = Function("y")
ode = Eq(-b**2*sin(x)**(2*m) - m*y(x)/tan(x) - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -b**2*sin(x)**(2*m) - m*y(x)/tan(x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method