2.5.13 Problem 14

Internal problem ID [13332]
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 : 14
Date solved : Wednesday, December 31, 2025 at 01:29:17 PM
CAS classification : [_Riccati]

2.5.13.1 Solved using first_order_ode_riccati

16.819 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=\sinh \left (\lambda x \right ) y^{2} a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \sinh \left (\lambda x \right ) y^{2} a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \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 \(f_0(x)=b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}\), \(f_1(x)=0\) and \(f_2(x)=\sinh \left (\lambda x \right ) a\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \sinh \left (\lambda x \right ) a} \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' &=\lambda \cosh \left (\lambda x \right ) a\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\sinh \left (\lambda x \right )^{3} a^{2} b \cosh \left (\lambda x \right )^{n} \end{align*}

Substituting the above terms back in equation (2) gives

\[ \sinh \left (\lambda x \right ) a u^{\prime \prime }\left (x \right )-\lambda \cosh \left (\lambda x \right ) a u^{\prime }\left (x \right )+\sinh \left (\lambda x \right )^{3} a^{2} b \cosh \left (\lambda x \right )^{n} u \left (x \right ) = 0 \]

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 \sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+c_2 \sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )} \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \sinh \left (\lambda x \right ) a} \\ y &= -\frac {\frac {c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )}}{\sinh \left (\lambda x \right ) a \left (c_1 \sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+c_2 \sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 \left (-\operatorname {BesselJ}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )}+\frac {c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) \lambda \sinh \left (\lambda x \right )}{2 \sqrt {\cosh \left (\lambda x \right )}}+\frac {2 c_3 \left (-\operatorname {BesselY}\left (\frac {1}{n +2}+1, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+\frac {\lambda \cosh \left (\lambda x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right ) \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )}\, \left (n +2\right )}}{\sinh \left (\lambda x \right ) a \left (\sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )+c_3 \sqrt {\cosh \left (\lambda x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= -\frac {\operatorname {sech}\left (\lambda x \right ) \left (-\cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right ) \sqrt {a}+\lambda \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= -\frac {\operatorname {sech}\left (\lambda x \right ) \left (-\cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right ) \sqrt {a}+\lambda \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (n +2\right )}\right )\right )} \\ \end{align*}

2.5.13.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 246

ode:=diff(y(x),x) = sinh(lambda*x)*y(x)^2*a+b*sinh(lambda*x)*cosh(lambda*x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\lambda \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )-a \left (\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}\right ) \operatorname {sech}\left (\lambda x \right )}{a^{{3}/{2}} \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )} \]

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 sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = lambda*cosh(lambda*x 
)/sinh(lambda*x)*diff(y(x),x)-a*sinh(lambda*x)^2*b*cosh(lambda*x)^n*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 symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying an equivalence, under non-integer power transformations, 
            to LODEs admitting Liouvillian solutions. 
            -> Trying a Liouvillian solution using Kovacics algorithm 
            <- No Liouvillian solutions exists 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            <- Bessel successful 
         <- special function solution successful 
         Change of variables used: 
            [x = arccosh(t)/lambda] 
         Linear ODE actually solved: 
            4*(t-1)^(1/2)*(t+1)^(1/2)*t^n*a*b*(t^2-1)*u(t)+4*(t-1)^(1/2)*(t+1)^\ 
(1/2)*lambda^2*(t^2-1)*diff(diff(u(t),t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \sinh \left (\lambda x \right ) y \left (x \right )^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{13332} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \sinh \left (\lambda x \right ) y \left (x \right )^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{13332} \end {array} \]

2.5.13.3 ✓ Mathematica. Time used: 0.457 (sec). Leaf size: 667

ode=D[y[x],x]==a*Sinh[\[Lambda]*x]*y[x]^2+b*Sinh[\[Lambda]*x]*Cosh[\[Lambda]*x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

2.5.13.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)**2*sinh(lambda_*x) - b*sinh(lambda_*x)*cosh(lambda_*x)**n + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(a*y(x)**2 + b*cosh(lambda_*x)**n)*sinh(lambda_*x) + Derivative(y(x), x) cannot be solved by the factorable group method