55.6.5 problem 22
Internal
problem
ID
[13366]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.4-2.
Equations
with
hyperbolic
tangent
and
cotangent.
Problem
number
:
22
Date
solved
:
Wednesday, October 01, 2025 at 08:37:40 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.057 (sec). Leaf size: 122
ode:=diff(y(x),x) = y(x)^2+a*lambda-a*(a+lambda)*coth(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\left (a +\lambda \right ) \coth \left (\lambda x \right ) \left (c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )\right )+c_1 \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right ) \lambda }{c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )}
\]
✓ Mathematica. Time used: 3.395 (sec). Leaf size: 340
ode=D[y[x],x]==y[x]^2+a*\[Lambda]-a*(a+\[Lambda])*Coth[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {a \int \frac {2 \lambda \left (e^{-2 \lambda x}+1\right )}{1-e^{2 \lambda x}} \, de^{2 \lambda x}}{2 \lambda }+\frac {2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {K[1] a+a+\lambda -\lambda K[1]}{\lambda (K[1]-1) K[1]}dK[1]+2 \lambda x\right )-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {K[1] a+a+\lambda -\lambda K[1]}{\lambda (K[1]-1) K[1]}dK[1]+4 \lambda x\right )+a \left (e^{2 \lambda x}+1\right ) \int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {K[1] a+a+\lambda -\lambda K[1]}{\lambda (K[1]-1) K[1]}dK[1]\right )dK[2]+a c_1 e^{2 \lambda x}+a c_1}{\left (e^{2 \lambda x}-1\right ) \left (\int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {K[1] a+a+\lambda -\lambda K[1]}{\lambda (K[1]-1) K[1]}dK[1]\right )dK[2]+c_1\right )}\\ y(x)&\to \frac {a \left (e^{2 \lambda x}+1\right )}{e^{2 \lambda x}-1}+\frac {a \int \frac {2 \lambda \left (e^{-2 \lambda x}+1\right )}{1-e^{2 \lambda x}} \, de^{2 \lambda x}}{2 \lambda } \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-a*lambda_ + a*(a + lambda_)/tanh(lambda_*x)**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2 + a**2/sinh(lambda_*x)**2 + a*lambda_/sinh(lambda_*x)**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method