2.6.9 Problem 26
Internal
problem
ID
[13344]
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
:
26
Date
solved
:
Sunday, January 18, 2026 at 07:25:15 PM
CAS
classification
:
[_Riccati]
2.6.9.1 Solved using first_order_ode_riccati
1.990 (sec)
Entering first order ode riccati solver
\begin{align*}
y^{\prime }&=y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}\), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u} \tag {1} \end{align*}
Using the above substitution in the given ODE results (after some simplification) in a second
order ODE to solve for \(u(x)\) which is
\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}
But
\begin{align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \end{align*}
Substituting the above terms back in equation (2) gives
\[
u^{\prime \prime }\left (x \right )+\left (-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}\right ) u \left (x \right ) = 0
\]
Unable to solve. Will ask Maple to solve
this ode now.
The solution for \(u \left (x \right )\) is
\begin{equation}
\tag{3} u \left (x \right ) = c_1 \operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right )+c_2 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = -2 c_1 \operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \lambda \coth \left (\lambda x \right )+c_1 \operatorname {csch}\left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right )+2 c_2 \sinh \left (\lambda x \right ) \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right ) \lambda \cosh \left (\lambda x \right )+c_2 \sinh \left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )+c_2 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-12 \coth \left (\lambda x \right )^{2} \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )+2 \lambda \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{2}-4 \coth \left (\lambda x \right )^{2} \operatorname {csch}\left (\lambda x \right )^{2} \lambda +\left (-\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )-1}+\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )+1}\right ) \operatorname {csch}\left (\lambda x \right )^{4}-4 \left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4} \lambda \coth \left (\lambda x \right )\right )
\end{equation}
Substituting equations (3,4) into (1) results in \begin{align*}
y &= \frac {-u'}{f_2 u} \\
y &= \frac {-u'}{u} \\
y &= -\frac {-2 c_1 \operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \lambda \coth \left (\lambda x \right )+c_1 \operatorname {csch}\left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right )+2 c_2 \sinh \left (\lambda x \right ) \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right ) \lambda \cosh \left (\lambda x \right )+c_2 \sinh \left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )+c_2 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-12 \coth \left (\lambda x \right )^{2} \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )+2 \lambda \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{2}-4 \coth \left (\lambda x \right )^{2} \operatorname {csch}\left (\lambda x \right )^{2} \lambda +\left (-\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )-1}+\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )+1}\right ) \operatorname {csch}\left (\lambda x \right )^{4}-4 \left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4} \lambda \coth \left (\lambda x \right )\right )}{c_1 \operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right )+c_2 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {-2 \operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \lambda \coth \left (\lambda x \right )+\operatorname {csch}\left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right )+2 c_3 \sinh \left (\lambda x \right ) \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right ) \lambda \cosh \left (\lambda x \right )+c_3 \sinh \left (\lambda x \right )^{2} \lambda \left (1-\tanh \left (\lambda x \right )^{2}\right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )+c_3 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-12 \coth \left (\lambda x \right )^{2} \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )+2 \lambda \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{2}-4 \coth \left (\lambda x \right )^{2} \operatorname {csch}\left (\lambda x \right )^{2} \lambda +\left (-\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )-1}+\frac {\lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )+1}\right ) \operatorname {csch}\left (\lambda x \right )^{4}-4 \left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4} \lambda \coth \left (\lambda x \right )\right )}{\operatorname {csch}\left (\lambda x \right )^{2} \tanh \left (\lambda x \right )+c_3 \sinh \left (\lambda x \right )^{2} \tanh \left (\lambda x \right ) \left (-4 \coth \left (\lambda x \right )^{3}+2 \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2}+\left (-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )\right ) \operatorname {csch}\left (\lambda x \right )^{4}\right )}
\]
Simplifying the above gives
\begin{align*}
y &= \frac {2 \left (-\frac {1}{2}+c_3 \left (-\cosh \left (\lambda x \right )^{2}+\frac {1}{2}\right ) \ln \left (\coth \left (\lambda x \right )-1\right )+\left (\cosh \left (\lambda x \right )^{2}-\frac {1}{2}\right ) c_3 \ln \left (\coth \left (\lambda x \right )+1\right )+4 \cosh \left (\lambda x \right )^{5} c_3 \sinh \left (\lambda x \right )-4 \cosh \left (\lambda x \right )^{3} c_3 \sinh \left (\lambda x \right )-\cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) c_3 +\cosh \left (\lambda x \right )^{2}\right ) \operatorname {sech}\left (\lambda x \right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{-4 \cosh \left (\lambda x \right )^{3} c_3 \sinh \left (\lambda x \right )+2 \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) c_3 +\ln \left (\coth \left (\lambda x \right )+1\right ) c_3 -\ln \left (\coth \left (\lambda x \right )-1\right ) c_3 +1} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {2 \left (-\frac {1}{2}+c_3 \left (-\cosh \left (\lambda x \right )^{2}+\frac {1}{2}\right ) \ln \left (\coth \left (\lambda x \right )-1\right )+\left (\cosh \left (\lambda x \right )^{2}-\frac {1}{2}\right ) c_3 \ln \left (\coth \left (\lambda x \right )+1\right )+4 \cosh \left (\lambda x \right )^{5} c_3 \sinh \left (\lambda x \right )-4 \cosh \left (\lambda x \right )^{3} c_3 \sinh \left (\lambda x \right )-\cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) c_3 +\cosh \left (\lambda x \right )^{2}\right ) \operatorname {sech}\left (\lambda x \right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{-4 \cosh \left (\lambda x \right )^{3} c_3 \sinh \left (\lambda x \right )+2 \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) c_3 +\ln \left (\coth \left (\lambda x \right )+1\right ) c_3 -\ln \left (\coth \left (\lambda x \right )-1\right ) c_3 +1} \\
\end{align*}
2.6.9.2 ✓ Maple. Time used: 0.008 (sec). Leaf size: 140
ode:=diff(y(x),x) = y(x)^2-2*lambda^2*tanh(lambda*x)^2-2*lambda^2*coth(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\lambda \left (c_1 \ln \left (\coth \left (\lambda x \right )-1\right ) \left (-2 \cosh \left (\lambda x \right )+\operatorname {sech}\left (\lambda x \right )\right )+c_1 \ln \left (\coth \left (\lambda x \right )+1\right ) \left (2 \cosh \left (\lambda x \right )-\operatorname {sech}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right ) \left (-1+4 \cosh \left (\lambda x \right )^{4}-4 \cosh \left (\lambda x \right )^{2}\right ) c_1 +2 \cosh \left (\lambda x \right )-\operatorname {sech}\left (\lambda x \right )\right )}{\sinh \left (\lambda x \right ) \left (4 \cosh \left (\lambda x \right )^{3} c_1 \sinh \left (\lambda x \right )-2 \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right ) c_1 -\ln \left (\coth \left (\lambda x \right )+1\right ) c_1 +\ln \left (\coth \left (\lambda x \right )-1\right ) c_1 -1\right )}
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying Riccati
trying Riccati Special
trying Riccati sub-methods:
trying Riccati_symmetries
trying Riccati to 2nd Order
-> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (2*lambda^2*tanh(
lambda*x)^2+2*lambda^2*coth(lambda*x)^2)*y(x), y(x)
*** Sublevel 2 ***
Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \
power @ Moebius
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\
(r(x), dx)) * 2F1([a1, a2], [b1], f)
-> Trying changes of variables to rationalize or make the ODE simpler
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Trying a Liouvillian solution using Kovacics algorithm
A Liouvillian solution exists
Reducible group (found an exponential solution)
Group is reducible, not completely reducible
<- Kovacics algorithm successful
Change of variables used:
[x = arccoth(t)/lambda]
Linear ODE actually solved:
(-2*t^4-2)*u(t)+(2*t^5-2*t^3)*diff(u(t),t)+(t^6-2*t^4+t^2)*diff(dif\
f(u(t),t),t) = 0
<- change of variables successful
<- Riccati to 2nd Order successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \end {array} \]
2.6.9.3 ✓ Mathematica. Time used: 3.547 (sec). Leaf size: 263
ode=D[y[x],x]==y[x]^2-2*\[Lambda]^2*Tanh[\[Lambda]*x]^2-2*\[Lambda]^2*Coth[\[Lambda]*x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2 \lambda \exp \left (-2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \left (\left (e^{4 \lambda x}+1\right ) \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right )+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]+4 \lambda x\right )+2 e^{4 \lambda x}-2 e^{8 \lambda x}\right )}{\left (e^{4 \lambda x}-1\right ) \left (\int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1\right )}\\ y(x)&\to \frac {2 \lambda \left (e^{4 \lambda x}+1\right )}{e^{4 \lambda x}-1} \end{align*}
2.6.9.4 ✗ Sympy
from sympy import *
x = symbols("x")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(2*lambda_**2*tanh(lambda_*x)**2 + 2*lambda_**2/tanh(lambda_*x)**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE 2*lambda_**2*tanh(lambda_*x)**2 + 2*lambda_**2/tanh(lambda_*x)**
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0
classify_ode(ode,func=y(x))
('lie_group',)