55.19.32 problem 32

Internal problem ID [13517]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 32
Date solved : Wednesday, October 01, 2025 at 03:51:38 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==f[x]*y[x]^2-a*Tan[\[Lambda]*x]^2*(a*f[x]-\[Lambda])+a*\[Lambda]; 
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") 
f = Function("f") 
ode = Eq(-a*lambda_ + a*(a*f(x) - lambda_)*tan(lambda_*x)**2 - f(x)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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