2.12.2 Problem 39

Internal problem ID [13401]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number : 39
Date solved : Friday, December 19, 2025 at 03:52:59 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \\ \end{align*}

Unknown ode type.

2.12.2.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 199

ode:=diff(y(x),x) = y(x)^2+lambda^2+3*a*lambda+a*(lambda-a)*cot(lambda*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\csc \left (\lambda x \right ) \left (-2 \operatorname {LegendreP}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda -2 \operatorname {LegendreQ}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_1 \lambda +\cos \left (\lambda x \right ) \left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_1 +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )\right )}{\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_1 +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )} \]

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 Riccati Special 
trying Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (a^2*cot(lambda*x)^2 
-a*cot(lambda*x)^2*lambda-3*a*lambda-lambda^2)*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\ 
(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      -> Trying changes of variables to rationalize or make the ODE simpler 
         trying a quadrature 
         checking if the LODE has constant coefficients 
         checking if the LODE is of Euler type 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a Liouvillian solution using Kovacics algorithm 
            A Liouvillian solution exists 
            Reducible group (found an exponential solution) 
            Group is reducible, not completely reducible 
            Solution has integrals. Trying a special function solution free of \ 
integrals... 
            -> Trying a solution in terms of special functions: 
               -> Bessel 
               -> elliptic 
               -> Legendre 
               <- Legendre successful 
            <- special function solution successful 
               -> Trying to convert hypergeometric functions to elementary form\ 
... 
               <- elementary form is not straightforward to achieve - returning\ 
 special function solution free of uncomputed integrals 
            <- Kovacics algorithm successful 
         Change of variables used: 
            [x = 1/lambda*arccos(t)] 
         Linear ODE actually solved: 
            (a^2*t^2+2*a*lambda*t^2+lambda^2*t^2-3*a*lambda-lambda^2)*u(t)+(-la\ 
mbda^2*t^3+lambda^2*t)*diff(u(t),t)+(-lambda^2*t^4+2*lambda^2*t^2-lambda^2)*dif\ 
f(diff(u(t),t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.12.2.2 ✓ Mathematica. Time used: 5.417 (sec). Leaf size: 306

ode=D[y[x],x]==y[x]^2+\[Lambda]^2+3*a*\[Lambda]+a*(\[Lambda]-a)*Cot[\[Lambda]*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

2.12.2.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-3*a*lambda_ - a*(-a + lambda_)/tan(lambda_*x)**2 - lambda_**2 - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a**2/tan(lambda_*x)**2 - 3*a*lambda_ - a*lambda_/tan(lambda_*x)**2 - lambda_**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method