61.6.8 problem 25

Internal problem ID [12070]
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 : 25
Date solved : Friday, March 14, 2025 at 04:00:06 AM
CAS classification : [_Riccati]

\begin{align*} \left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 302

ode:=(a*coth(lambda*x)+b)*diff(y(x),x) = y(x)^2+c*coth(x*mu)*y(x)-d^2+c*d*coth(x*mu); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-{\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \coth \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \coth \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \coth \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (\lambda x \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (\lambda x \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}d x \right )}d x -c_{1}} \]

✓ Mathematica. Time used: 38.678 (sec). Leaf size: 808

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
cg = symbols("cg") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(-c*d/tanh(mu*x) - c*y(x)/tanh(mu*x) + d**2 + (a/tanh(cg*x) + b)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Invalid NaN comparison