61.12.2 problem 39
Internal
problem
ID
[12134]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-4.
Equations
with
cotangent.
Problem
number
:
39
Date
solved
:
Wednesday, March 05, 2025 at 04:41:50 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 199
ode:=diff(y(x),x) = y(x)^2+lambda^2+3*a*lambda+a*(-a+lambda)*cot(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\csc \left (\lambda x \right ) \left (-2 \operatorname {LegendreP}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda -2 \operatorname {LegendreQ}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda +\cos \left (\lambda x \right ) \left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )\right )}{\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )}
\]
✓ Mathematica. Time used: 6.62 (sec). Leaf size: 306
ode=D[y[x],x]==y[x]^2+\[Lambda]^2+3*a*\[Lambda]+a*(\[Lambda]-a)*Cot[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda x) e^{-\text {arctanh}(\cos (2 \lambda x))} \left (c_1 \sin ^{\frac {a}{\lambda }}(2 \lambda x) ((a+\lambda ) \cos (2 \lambda x)+a-\lambda ) e^{\text {arctanh}(\cos (2 \lambda x))} \int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]+\sin ^{\frac {a}{\lambda }}(2 \lambda x) ((a+\lambda ) \cos (2 \lambda x)+a-\lambda ) e^{\text {arctanh}(\cos (2 \lambda x))}+c_1 e^{\frac {a \text {arctanh}(\cos (2 \lambda x))}{\lambda }}\right )}{1+c_1 \int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]} \\
y(x)\to \csc (2 \lambda x) \left (-\frac {\sin ^{-\frac {a}{\lambda }}(2 \lambda x) e^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda x))}{\lambda }}}{\int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]}-(a+\lambda ) \cos (2 \lambda x)-a+\lambda \right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
cg = symbols("cg")
y = Function("y")
ode = Eq(-3*a*cg - a*(-a + cg)/tan(cg*x)**2 - cg**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2/tan(cg*x)**2 - 3*a*cg - a*cg/tan(cg*x)**2 - cg**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method