61.13.8 problem 54
Internal
problem
ID
[12149]
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
:
54
Date
solved
:
Wednesday, March 05, 2025 at 04:55:32 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 226
ode:=diff(y(x),x) = y(x)^2-m*y(x)*tan(x)+b^2*cos(x)^(2*m);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-\cos \left (x \right ) \sqrt {\cos \left (x \right )^{2 m -2}}\, \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) c_{1} +\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\right ) b \cos \left (x \right )^{-m +1} \left (\left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right ) \sin \left (x \right )^{2}-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )}{3 c_{1} \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )}
\]
✓ Mathematica. Time used: 2.588 (sec). Leaf size: 73
ode=D[y[x],x]==y[x]^2-m*y[x]*Tan[x]+b^2*Cos[x]^(2*m);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \sqrt {b^2} \cos ^m(x) \tan \left (-\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(x)\right )}{m+1}+c_1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
b = symbols("b")
m = symbols("m")
y = Function("y")
ode = Eq(-b**2*cos(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*cos(x)**(2*m) + m*y(x)*tan(x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method