2.19.32 Problem 33

Internal problem ID [13480]
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 : 33
Date solved : Friday, December 19, 2025 at 04:53:44 AM
CAS classification : [_Riccati]

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

Unknown ode type.

2.19.32.1 ✗ Maple

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

\[ \text {No solution found} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying inverse_Riccati 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
   -> Computing symmetries using: way = 6 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
   -> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE, diff(y(x),x) = 0, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = 2*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+diff(f(x),x)*y(x)/f(x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+cot(lambda*x)*y(x)*(-2*f(x)* 
cot(lambda*x)^2*a*lambda+2*cot(lambda*x)^2*lambda^2+diff(f(x),x)*cot(lambda*x)* 
a-2*f(x)*a*lambda+2*lambda^2)/(f(x)*cot(lambda*x)^2*a-cot(lambda*x)^2*lambda- 
lambda), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-y(x)*cot(lambda*x)*(2*f(x)*cot 
(lambda*x)^2*a*lambda-2*cot(lambda*x)^2*lambda^2-diff(f(x),x)*cot(lambda*x)*a+2 
*f(x)*a*lambda-2*lambda^2)/(f(x)*cot(lambda*x)^2*a-cot(lambda*x)^2*lambda- 
lambda), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = -diff(f(x),x)*y(x)/f(x), y(x 
) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-2*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}-a \cot \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}-a \cot \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \end {array} \]

2.19.32.2 ✗ Mathematica

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

Not solved

2.19.32.3 ✗ 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_ - a*lambda_/tan(lambda_*x)**2 - f(x)*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method