55.6.9 problem 26

Internal problem ID [13370]
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 : 26
Date solved : Wednesday, October 01, 2025 at 08:45:21 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 140

ode:=diff(y(x),x) = y(x)^2-2*lambda^2*tanh(lambda*x)^2-2*lambda^2*coth(lambda*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\lambda \left (c_1 \ln \left (\coth \left (\lambda x \right )-1\right ) \left (-2 \cosh \left (\lambda x \right )+\operatorname {sech}\left (\lambda x \right )\right )+c_1 \ln \left (\coth \left (\lambda x \right )+1\right ) \left (2 \cosh \left (\lambda x \right )-\operatorname {sech}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right ) \left (-1+4 \cosh \left (\lambda x \right )^{4}-4 \cosh \left (\lambda x \right )^{2}\right ) c_1 +2 \cosh \left (\lambda x \right )-\operatorname {sech}\left (\lambda x \right )\right )}{\sinh \left (\lambda x \right ) \left (4 \cosh \left (\lambda x \right )^{3} c_1 \sinh \left (\lambda x \right )-2 \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) c_1 +\ln \left (\coth \left (\lambda x \right )-1\right ) c_1 -\ln \left (\coth \left (\lambda x \right )+1\right ) c_1 -1\right )} \]

✓ Mathematica. Time used: 3.547 (sec). Leaf size: 263

ode=D[y[x],x]==y[x]^2-2*\[Lambda]^2*Tanh[\[Lambda]*x]^2-2*\[Lambda]^2*Coth[\[Lambda]*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 \lambda \exp \left (-2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \left (\left (e^{4 \lambda x}+1\right ) \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right ) \int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]\right )+c_1 \exp \left (2 \int _1^{e^{4 x \lambda }}\frac {1}{K[1]-K[1]^2}dK[1]+4 \lambda x\right )+2 e^{4 \lambda x}-2 e^{8 \lambda x}\right )}{\left (e^{4 \lambda x}-1\right ) \left (\int _1^{e^{4 x \lambda }}\exp \left (-2 \int _1^{K[2]}\frac {1}{K[1]-K[1]^2}dK[1]\right )dK[2]+c_1\right )}\\ y(x)&\to \frac {2 \lambda \left (e^{4 \lambda x}+1\right )}{e^{4 \lambda x}-1} \end{align*}

✗ Sympy

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

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