61.13.3 problem 49
Internal
problem
ID
[12144]
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
:
49
Date
solved
:
Wednesday, March 05, 2025 at 04:52:26 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+a \cos \left (\lambda x \right )^{n} y-a \cos \left (\lambda x \right )^{n -1} \end{align*}
✗ Maple
ode:=diff(y(x),x) = lambda*sin(lambda*x)*y(x)^2+a*cos(lambda*x)^n*y(x)-a*cos(lambda*x)^(n-1);
dsolve(ode,y(x), singsol=all);
\[ \text {No solution found} \]
✓ Mathematica. Time used: 20.858 (sec). Leaf size: 467
ode=D[y[x],x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+a*Cos[\[Lambda]*x]^n*y[x]-a*Cos[\[Lambda]*x]^(n-1);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \tan (\lambda K[1]) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) y(x) \cos ^{n+1}(\lambda K[1])+\lambda y(x)^2 \cos (\lambda K[1])\right )}{(\cos (\lambda K[1]) y(x)-1)^2}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (-\frac {a \cos ^{n+1}(x \lambda ) \csc (x \lambda ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(x \lambda )\right ) \sqrt {\sin ^2(x \lambda )}}{(n+1) \lambda }\right )}{(\cos (x \lambda ) K[2]-1)^2}-\int _1^x\left (\frac {2 \exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) K[2] \cos ^{n+1}(\lambda K[1])+\lambda K[2]^2 \cos (\lambda K[1])\right ) \sin (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^3}-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (a \csc (\lambda K[1]) \cos ^{n+1}(\lambda K[1])+2 \lambda K[2] \cos (\lambda K[1])\right ) \tan (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
cg = symbols("cg")
n = symbols("n")
y = Function("y")
ode = Eq(-a*y(x)*cos(cg*x)**n + a*cos(cg*x)**(n - 1) - cg*y(x)**2*sin(cg*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*y(x)*cos(cg*x)**n + a*cos(cg*x)**(n - 1) - cg*y(x)**2*sin(cg*x) + Derivative(y(x), x) cannot be solved by the lie group method