2.5.9 Problem 9
Internal
problem
ID
[13328]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.4-1.
Equations
with
hyperbolic
sine
and
cosine
Problem
number
:
9
Date
solved
:
Wednesday, December 31, 2025 at 01:25:47 PM
CAS
classification
:
[_Riccati]
2.5.9.1 Solved using first_order_ode_riccati
18.420 (sec)
Entering first order ode riccati solver
\begin{align*}
y^{\prime }&=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = \textit {the\_rhs}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=a b \cosh \left (\beta x \right )-b^{2}\), \(f_1(x)=\cosh \left (\beta x \right ) a\) 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 &=\cosh \left (\beta x \right ) a\\ f_2^2 f_0 &=a b \cosh \left (\beta x \right )-b^{2} \end{align*}
Substituting the above terms back in equation (2) gives
\[
u^{\prime \prime }\left (x \right )-\cosh \left (\beta x \right ) a u^{\prime }\left (x \right )+\left (a b \cosh \left (\beta x \right )-b^{2}\right ) u \left (x \right ) = 0
\]
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 {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+c_2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )}{2}+\frac {c_2 \beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}{2}+c_2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \left (a \cosh \left (\frac {\beta x}{2}\right )^{2}+a \sinh \left (\frac {\beta x}{2}\right )^{2}\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}
\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 \beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )}{2}+\frac {c_2 \beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}{2}+c_2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \left (a \cosh \left (\frac {\beta x}{2}\right )^{2}+a \sinh \left (\frac {\beta x}{2}\right )^{2}\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}{c_1 \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+c_2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {\frac {\beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )}{2}+\frac {c_3 \beta \left (1-\coth \left (\frac {\beta x}{2}\right )^{2}\right ) \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}{2}+c_3 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \left (a \cosh \left (\frac {\beta x}{2}\right )^{2}+a \sinh \left (\frac {\beta x}{2}\right )^{2}\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}{\operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+c_3 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {2 a \sinh \left (\frac {\beta x}{2}\right ) \cosh \left (\frac {\beta x}{2}\right )}{\beta }}}
\]
Simplifying the above gives
\begin{align*}
y &= \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_3 a \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_3 +\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )\right )}{2 c_3 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_3 a \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_3 +\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )\right )}{2 c_3 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \\
\end{align*}
2.5.9.2 Solved using first_order_ode_riccati_by_guessing_particular_solution
0.188 (sec)
Entering first order ode riccati guess solver
\begin{align*}
y^{\prime }&=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \\
\end{align*}
This is a Riccati ODE. Comparing the above ODE to
solve with the Riccati standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows that \begin{align*} f_0(x) & =a b \cosh \left (\beta x \right )-b^{2}\\ f_1(x) & =\cosh \left (\beta x \right ) a\\ f_2(x) &=1 \end{align*}
Using trial and error, the following particular solution was found
\[
y_p = -b
\]
Since a particular solution is
known, then the general solution is given by \begin{align*} y &= y_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}
Where
\begin{align*} \phi (x) &= e^{ \int 2 f_2 y_p + f_1 \,dx } \end{align*}
Evaluating the above gives the general solution as
\[
y = -b +\frac {{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}}{c_1 -\int {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}d x}
\]
Summary of solutions found
\begin{align*}
y &= -b +\frac {{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}}{c_1 -\int {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}d x} \\
\end{align*}
2.5.9.3 ✓ Maple. Time used: 0.003 (sec). Leaf size: 73
ode:=diff(y(x),x) = y(x)^2+a*cosh(beta*x)*y(x)+a*b*cosh(beta*x)-b^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {b \int {\mathrm e}^{\frac {-2 x b \beta +\sinh \left (\beta x \right ) a}{\beta }}d x -b c_1 +{\mathrm e}^{\frac {-2 x b \beta +\sinh \left (\beta x \right ) a}{\beta }}}{-\int {\mathrm e}^{\frac {-2 x b \beta +\sinh \left (\beta x \right ) a}{\beta }}d x +c_1}
\]
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 sub-methods:
<- Riccati particular case Kamke (b) successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \cosh \left (\beta x \right ) y \left (x \right )+a b \cosh \left (\beta x \right )-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \cosh \left (\beta x \right ) y \left (x \right )+a b \cosh \left (\beta x \right )-b^{2} \end {array} \]
2.5.9.4 ✓ Mathematica. Time used: 2.703 (sec). Leaf size: 298
ode=D[y[x],x]==y[x]^2+a*Cosh[\[Beta]*x]*y[x]+a*b*Cosh[\[Beta]*x]-b^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )+b \int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+b c_1}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+c_1}\\ y(x)&\to -b\\ y(x)&\to -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]}-b \end{align*}
2.5.9.5 ✗ Sympy
from sympy import *
x = symbols("x")
BETA = symbols("BETA")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*b*cosh(BETA*x) - a*y(x)*cosh(BETA*x) + b**2 - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*b*cosh(BETA*x) - a*y(x)*cosh(BETA*x) + b**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method