61.6.3 problem 20

Internal problem ID [12065]
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 : 20
Date solved : Sunday, March 30, 2025 at 10:30:13 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 85

ode:=diff(y(x),x) = y(x)^2+a*x*tanh(b*x)^m*y(x)+a*tanh(b*x)^m; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-x \,{\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x}-\int {\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x}d x +c_1}{\left (-c_1 +\int {\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]

✓ Mathematica. Time used: 2.44 (sec). Leaf size: 126

ode=D[y[x],x]==y[x]^2+a*x*Tanh[b*x]^m*y[x]+a*Tanh[b*x]^m; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-a K[1] \tanh ^m(b K[1])dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \tanh ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \tanh ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}

✗ Sympy

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

Timed Out