61.5.11 problem 11

Internal problem ID [12056]
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 : 11
Date solved : Sunday, March 30, 2025 at 10:24:50 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 104

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

\[ y = \frac {2 \lambda \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {sech}\left (\lambda x \right )^{2}\right )d x c_1 \tanh \left (\lambda x \right )+2 \,{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} c_1 \lambda \operatorname {sech}\left (\lambda x \right )^{2}-\tanh \left (\lambda x \right )}{2 \lambda \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {sech}\left (\lambda x \right )^{2}\right )d x c_1 -1} \]

✓ Mathematica. Time used: 16.097 (sec). Leaf size: 211

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

\begin{align*} y(x)\to \frac {\text {sech}^2(\lambda x) \left (c_1 \sinh (2 \lambda x) \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]+2 c_1 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}+\sinh (2 \lambda x)\right )}{2+2 c_1 \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]} \\ y(x)\to \frac {1}{2} \text {sech}^2(\lambda x) \left (\frac {2 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}}{\int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]}+\sinh (2 \lambda x)\right ) \\ \end{align*}

✗ Sympy

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

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