2.6.10 Problem 27
Internal
problem
ID
[13345]
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
:
27
Date
solved
:
Sunday, January 18, 2026 at 07:25:20 PM
CAS
classification
:
[_Riccati]
2.6.10.1 Solved using first_order_ode_riccati
15.190 (sec)
Entering first order ode riccati solver
\begin{align*}
y^{\prime }&=y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= -\tanh \left (\lambda x \right )^{2} a^{2}-\tanh \left (\lambda x \right )^{2} a \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda +y^{2} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = -\tanh \left (\lambda x \right )^{2} a^{2}-\tanh \left (\lambda x \right )^{2} a \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda +y^{2}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=-\tanh \left (\lambda x \right )^{2} a^{2}-\tanh \left (\lambda x \right )^{2} a \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda \), \(f_1(x)=0\) 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 &=-\tanh \left (\lambda x \right )^{2} a^{2}-\tanh \left (\lambda x \right )^{2} a \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda \end{align*}
Substituting the above terms back in equation (2) gives
\[
u^{\prime \prime }\left (x \right )+\left (-\tanh \left (\lambda x \right )^{2} a^{2}-\tanh \left (\lambda x \right )^{2} a \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda \right ) u \left (x \right ) = 0
\]
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 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = -\frac {c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} a \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}-c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+2 c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }+1\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \left (1-\coth \left (\lambda x \right )^{2}\right )-c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\frac {c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (a +\lambda \right ) \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}+\frac {2 c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\lambda -2 b \right ) \operatorname {hypergeom}\left (\left [2, \frac {\lambda -2 b}{2 \lambda }+1\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }+1\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{3 \lambda +2 a}
\end{equation}
Substituting equations (3,4) into (1) results in \begin{align*}
y &= \frac {-u'}{f_2 u} \\
y &= \frac {-u'}{u} \\
y &= -\frac {-\frac {c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} a \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}-c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+2 c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }+1\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \left (1-\coth \left (\lambda x \right )^{2}\right )-c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\frac {c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (a +\lambda \right ) \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}+\frac {2 c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\lambda -2 b \right ) \operatorname {hypergeom}\left (\left [2, \frac {\lambda -2 b}{2 \lambda }+1\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }+1\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{3 \lambda +2 a}}{c_1 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+c_2 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {-\frac {\coth \left (\lambda x \right )^{-\frac {a}{\lambda }} a \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}-\coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+2 \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }+1\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \left (1-\coth \left (\lambda x \right )^{2}\right )-c_3 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \left (-b -a \right ) \coth \left (\lambda x \right ) \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\frac {c_3 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (a +\lambda \right ) \left (1-\coth \left (\lambda x \right )^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}{\coth \left (\lambda x \right )}+\frac {2 c_3 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\lambda -2 b \right ) \operatorname {hypergeom}\left (\left [2, \frac {\lambda -2 b}{2 \lambda }+1\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }+1\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right ) \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{3 \lambda +2 a}}{\coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \operatorname {hypergeom}\left (\left [\frac {-b -a}{\lambda }\right ], \left [\right ], \coth \left (\lambda x \right )^{2}\right )+c_3 \operatorname {csch}\left (\lambda x \right )^{\frac {-b -a}{\lambda }} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}
\]
Simplifying the above gives
\begin{align*}
y &= \frac {-4 \lambda \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{2} c_3 \left (b -\frac {\lambda }{2}\right ) \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {5 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (\frac {3 \lambda }{2}+a \right ) \left (c_3 \left (\left (-b -a \right ) \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }}+\coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \operatorname {csch}\left (\lambda x \right ) \operatorname {sech}\left (\lambda x \right ) \left (a +\lambda \right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }} \left (a \tanh \left (\lambda x \right )+\coth \left (\lambda x \right ) b \right )\right )}{\left (3 \lambda +2 a \right ) \left (\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}+\operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) c_3 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}\right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {-4 \lambda \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{2} c_3 \left (b -\frac {\lambda }{2}\right ) \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {5 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (\frac {3 \lambda }{2}+a \right ) \left (c_3 \left (\left (-b -a \right ) \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }}+\coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \operatorname {csch}\left (\lambda x \right ) \operatorname {sech}\left (\lambda x \right ) \left (a +\lambda \right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }} \left (a \tanh \left (\lambda x \right )+\coth \left (\lambda x \right ) b \right )\right )}{\left (3 \lambda +2 a \right ) \left (\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}+\operatorname {hypergeom}\left (\left [1, \frac {\lambda -2 b}{2 \lambda }\right ], \left [\frac {3 \lambda +2 a}{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) c_3 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}\right )} \\
\end{align*}
2.6.10.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 289
ode:=diff(y(x),x) = y(x)^2+lambda*a+b*lambda-2*a*b-a*(a+lambda)*tanh(lambda*x)^2-b*(b+lambda)*coth(lambda*x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-4 \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }} \operatorname {csch}\left (\lambda x \right )^{2} \lambda \left (b -\frac {\lambda }{2}\right ) c_1 \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )-2 \left (\left (\left (\frac {3 a}{2}+\frac {3 b}{2}\right ) \lambda +a b \right ) \coth \left (\lambda x \right )^{\frac {2 \lambda +2 a}{\lambda }}+\coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \lambda \left (a +\frac {3 \lambda }{5}\right ) \operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right )\right ) c_1 \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (a +\frac {3 \lambda }{2}\right ) \left (\tanh \left (\lambda x \right ) a +b \coth \left (\lambda x \right )\right ) \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (2 a +3 \lambda \right ) \left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) c_1 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+\left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right )}
\]
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) = (a^2*tanh(lambda*x)^
2+a*tanh(lambda*x)^2*lambda+b^2*coth(lambda*x)^2+b*coth(lambda*x)^2*lambda+2*a*
b-a*lambda-b*lambda)*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
Solution has integrals. Trying a special function solution free of \
integrals...
-> Trying a solution in terms of special functions:
-> Bessel
-> elliptic
-> Legendre
-> Whittaker
-> hyper3: Equivalence to 1F1 under a power @ Moebius
-> hypergeometric
-> heuristic approach
<- heuristic approach successful
<- hypergeometric successful
<- special function solution successful
-> Trying to convert hypergeometric functions to elementary form\
...
<- elementary form could result into a too large expression - re\
turning special function form of solution, free of uncomputed integrals
<- Kovacics algorithm successful
Change of variables used:
[x = arccoth(t)/lambda]
Linear ODE actually solved:
(-b^2*t^4-b*lambda*t^4-2*a*b*t^2+a*lambda*t^2+b*lambda*t^2-a^2-a*la\
mbda)*u(t)+(2*lambda^2*t^5-2*lambda^2*t^3)*diff(u(t),t)+(lambda^2*t^6-2*lambda^\
2*t^4+lambda^2*t^2)*diff(diff(u(t),t),t) = 0
<- change of variables successful
<- Riccati to 2nd Order successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \end {array} \]
2.6.10.3 ✓ Mathematica. Time used: 16.354 (sec). Leaf size: 583
ode=D[y[x],x]==y[x]^2+a*\[Lambda]+b*\[Lambda]-2*a*b-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2-b*(b+\[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}+4 e^{4 \lambda x}-6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )+b \left (-2 e^{2 \lambda x}+4 e^{4 \lambda x}+6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )\right )}{\left (e^{4 \lambda x}-1\right )^2} \, de^{2 \lambda x}}{2 \lambda }+\frac {2 \lambda \exp \left (\int _1^{e^{2 x \lambda }}\frac {a (K[1]-1)^2+(K[1]+1) (K[1] b+b+\lambda -\lambda 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+(K[1]+1) (K[1] b+b+\lambda -\lambda 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+b \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+(K[1]+1) (K[1] b+b+\lambda -\lambda K[1])}{\lambda K[1] \left (K[1]^2-1\right )}dK[1]\right )dK[2]-2 c_1 (a-b) e^{2 \lambda x}+c_1 (a+b) e^{4 \lambda x}+c_1 (a+b)}{\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+(K[1]+1) (K[1] b+b+\lambda -\lambda 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+b \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}+4 e^{4 \lambda x}-6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )+b \left (-2 e^{2 \lambda x}+4 e^{4 \lambda x}+6 e^{6 \lambda x}+e^{8 \lambda x}-1\right )\right )}{\left (e^{4 \lambda x}-1\right )^2} \, de^{2 \lambda x}}{2 \lambda } \end{align*}
2.6.10.4 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(2*a*b - a*lambda_ + a*(a + lambda_)*tanh(lambda_*x)**2 - b*lambda_ + b*(b + lambda_)/tanh(lambda_*x)**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0