61.6.6 problem 23
Internal
problem
ID
[12068]
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
:
23
Date
solved
:
Sunday, March 30, 2025 at 10:33:40 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \end{align*}
✓ Maple. Time used: 0.076 (sec). Leaf size: 148
ode:=diff(y(x),x) = y(x)^2-lambda^2+3*a*lambda-a*(a+lambda)*coth(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\left (a +\lambda \right ) \coth \left (\lambda x \right ) \left (c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )\right )+2 c_1 \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right ) \lambda }{c_1 \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}
\]
✓ Mathematica. Time used: 6.838 (sec). Leaf size: 496
ode=D[y[x],x]==y[x]^2-\[Lambda]^2+3*a*\[Lambda]-a*(a+\[Lambda])*Coth[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (a \left (2 e^{2 \lambda x}-e^{4 \lambda x}+1\right )+\lambda \left (2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )\right )}{e^{4 \lambda x}-1} \, de^{2 \lambda x}}{2 \lambda }+\frac {2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]+1)^2-2 \lambda (K[1]-1) K[1]}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]+2 \lambda x\right )-2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]+1)^2-2 \lambda (K[1]-1) K[1]}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]+6 \lambda x\right )+\left (a \left (e^{2 \lambda x}+1\right )^2-\lambda \left (e^{2 \lambda x}-1\right )^2\right ) \int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {a (K[1]+1)^2-2 \lambda (K[1]-1) K[1]}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1 (a-\lambda )-c_1 (\lambda -a) e^{4 \lambda x}+2 c_1 (a+\lambda ) e^{2 \lambda x}}{\left (e^{4 \lambda x}-1\right ) \left (\int _1^{e^{2 x \lambda }}\exp \left (\int _1^{K[2]}\frac {a (K[1]+1)^2-2 \lambda (K[1]-1) K[1]}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1\right )} \\
y(x)\to \frac {a \left (e^{2 \lambda x}+1\right )^2-\lambda \left (e^{2 \lambda x}-1\right )^2}{e^{4 \lambda x}-1}-\frac {\int \frac {2 \lambda e^{-2 \lambda x} \left (a \left (2 e^{2 \lambda x}-e^{4 \lambda x}+1\right )+\lambda \left (2 e^{2 \lambda x}+e^{4 \lambda x}-1\right )\right )}{e^{4 \lambda x}-1} \, 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(-3*a*lambda_ + a*(a + lambda_)/tanh(lambda_*x)**2 + lambda_**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2/tanh(lambda_*x)**2 - 3*a*lambda_ + a*lambda_/tanh(lambda_*x)**2 + lambda_**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method