61.13.13 problem 59

Internal problem ID [12154]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing combinations of trigonometric functions.
Problem number : 59
Date solved : Sunday, March 30, 2025 at 11:22:18 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.157 (sec). Leaf size: 61

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

\[ y = \frac {\operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) c_1 a +1}{\cos \left (\lambda x \right ) \operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) c_1 a -{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{\lambda }} c_1 \lambda +\cos \left (\lambda x \right )} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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