2.5.7 Problem 7

Internal problem ID [13326]
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 : 7
Date solved : Wednesday, December 31, 2025 at 01:24:15 PM
CAS classification : [_Riccati]

2.5.7.1 Solved using first_order_ode_riccati

9.112 (sec)

Entering first order ode riccati solver

\begin{align*} \left (\sinh \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )+\lambda ^{2} \sinh \left (\lambda x \right ) a&=0 \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {\sinh \left (\lambda x \right ) y^{2} a -\lambda ^{2} \sinh \left (\lambda x \right ) a +y^{2} b}{\sinh \left (\lambda x \right ) a +b} \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 \(f_0(x)=-\frac {\lambda ^{2} \sinh \left (\lambda x \right ) a}{\sinh \left (\lambda x \right ) a +b}\), \(f_1(x)=0\) 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 &=-\frac {\lambda ^{2} \sinh \left (\lambda x \right ) a}{\sinh \left (\lambda x \right ) a +b} \end{align*}

Substituting the above terms back in equation (2) gives

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

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 ) = \frac {\left (\tanh \left (\frac {\lambda x}{2}\right )^{2} \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{3}+\tanh \left (\frac {\lambda x}{2}\right )^{2} \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) b^{5}-2 \tanh \left (\frac {\lambda x}{2}\right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b^{2}-2 \tanh \left (\frac {\lambda x}{2}\right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{4}+\tanh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} a^{2}-\operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{3}-\operatorname {arctanh}\left (\frac {b \tanh \left (\frac {\lambda x}{2}\right )-a}{\sqrt {a^{2}+b^{2}}}\right ) b^{5}+\left (a^{2}+b^{2}\right )^{{3}/{2}} a b \right ) \left (\sinh \left (\lambda x \right ) a +b \right ) c_1}{\tanh \left (\frac {\lambda x}{2}\right )^{2} b -2 a \tanh \left (\frac {\lambda x}{2}\right )-b}+c_2 \left (\sinh \left (\lambda x \right ) a +b \right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} \text {Expression too large to display} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u} \\ y &= \text {Expression too large to display} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ \text {Expression too large to display} \]

Simplifying the above gives

\begin{align*} y &= -\frac {2 \left (\left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )-c_3 \right ) a \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+a \sinh \left (\frac {\lambda x}{2}\right ) b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (a \cosh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+\cosh \left (\frac {\lambda x}{2}\right ) b \right )+2 \left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )-c_3 \right ) \left (a \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= -\frac {2 \left (\left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )-c_3 \right ) a \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+a \sinh \left (\frac {\lambda x}{2}\right ) b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (a \cosh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+\cosh \left (\frac {\lambda x}{2}\right ) b \right )+2 \left (a^{2} b^{2} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )+b^{4} \operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right )-c_3 \right ) \left (a \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \\ \end{align*}

2.5.7.2 ✓ Maple. Time used: 0.005 (sec). Leaf size: 250

ode:=(sinh(lambda*x)*a+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {4 \left (a \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_1 \right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+a b \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {\lambda x}{2}\right ) \left (a^{2}+b^{2}\right )^{{3}/{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (a \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {\lambda x}{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_1 \right )\right )} \]

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) = 1/(a*sinh(lambda*x)+ 
b)*a*lambda^2*sinh(lambda*x)*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)] 
      <- linear_1 successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \sinh \left (\lambda x \right )+b \right ) \left (\frac {d}{d x}y \left (x \right )-y \left (x \right )^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\sinh \left (\lambda x \right ) y \left (x \right )^{2} a -a \,\lambda ^{2} \sinh \left (\lambda x \right )+y \left (x \right )^{2} b}{a \sinh \left (\lambda x \right )+b} \end {array} \]

2.5.7.3 ✓ Mathematica. Time used: 11.48 (sec). Leaf size: 229

ode=(a*Sinh[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Sinh[\[Lambda]*x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x-\frac {-a \sinh (\lambda K[1]) \lambda ^2+b y(x)^2+a \sinh (\lambda K[1]) y(x)^2}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b y(x)+a \sinh (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(a \lambda \cosh (x \lambda )+b K[2]+a K[2] \sinh (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (-a \sinh (\lambda K[1]) \lambda ^2+b K[2]^2+a K[2]^2 \sinh (\lambda K[1])\right )}{(a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \sinh (\lambda K[1]) K[2]}{(b+a \sinh (\lambda K[1])) (a \lambda \cosh (\lambda K[1])+b K[2]+a K[2] \sinh (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

2.5.7.4 ✗ Sympy

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

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