2.11.8 Problem 34

Internal problem ID [13396]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with tangent.
Problem number : 34
Date solved : Wednesday, December 31, 2025 at 02:22:26 PM
CAS classification : [_Riccati]

2.11.8.1 Solved using first_order_ode_riccati

65.366 (sec)

Entering first order ode riccati solver

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

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \textit {the\_rhs} \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=-a \,b^{2} \tan \left (\lambda x \right )^{n} \tan \left (\lambda x \right )^{2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \), \(f_1(x)=0\) and \(f_2(x)=\tan \left (\lambda x \right )^{n} a\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \tan \left (\lambda x \right )^{n} a} \tag {1} \end{align*}

Using the above substitution in the given ODE results (after some simplification) in a second order ODE to solve for \(u(x)\) which is

\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}

But

\begin{align*} f_2' &=\frac {\tan \left (\lambda x \right )^{n} n \lambda \left (1+\tan \left (\lambda x \right )^{2}\right ) a}{\tan \left (\lambda x \right )}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\tan \left (\lambda x \right )^{2 n} a^{2} \left (-a \,b^{2} \tan \left (\lambda x \right )^{n} \tan \left (\lambda x \right )^{2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ \tan \left (\lambda x \right )^{n} a u^{\prime \prime }\left (x \right )-\frac {\tan \left (\lambda x \right )^{n} n \lambda \left (1+\tan \left (\lambda x \right )^{2}\right ) a u^{\prime }\left (x \right )}{\tan \left (\lambda x \right )}+\tan \left (\lambda x \right )^{2 n} a^{2} \left (-a \,b^{2} \tan \left (\lambda x \right )^{n} \tan \left (\lambda x \right )^{2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \right ) u \left (x \right ) = 0 \]

Unable to solve. Will ask Maple to solve this ode now.

The solution for \(u \left (x \right )\) is

\begin{equation} \tag{3} u \left (x \right ) = {\mathrm e}^{\int -\frac {\tan \left (\lambda x \right )^{n +1} a \left (\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x b +c_1 b +{\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}\right )}{\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x +c_1}d x} c_2 \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -\frac {\tan \left (\lambda x \right )^{n +1} a \left (\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x b +c_1 b +{\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}\right ) {\mathrm e}^{\int -\frac {\tan \left (\lambda x \right )^{n +1} a \left (\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x b +c_1 b +{\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}\right )}{\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x +c_1}d x} c_2}{\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x +c_1} \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \tan \left (\lambda x \right )^{n} a} \\ y &= \frac {\tan \left (\lambda x \right )^{n +1} \left (\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x b +c_1 b +{\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}\right ) \tan \left (\lambda x \right )^{-n}}{\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x +c_1} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {\tan \left (\lambda x \right )^{n +1} \left (\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x b +c_1 b +{\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}\right ) \tan \left (\lambda x \right )^{-n}}{\int -\cot \left (\lambda x \right ) a \tan \left (\lambda x \right )^{n +2} {\mathrm e}^{-\int \cot \left (\lambda x \right ) \left (-2 \tan \left (\lambda x \right )^{n +2} a b +\lambda \tan \left (\lambda x \right )^{2}+\lambda \right )d x}d x +c_1} \\ \end{align*}

2.11.8.2 ✗ Maple

ode:=diff(y(x),x) = a*tan(lambda*x)^n*y(x)^2-a*b^2*tan(lambda*x)^(n+2)+b*lambda*tan(lambda*x)^2+b*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 Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = n*lambda*(1+tan( 
lambda*x)^2)/tan(lambda*x)*diff(y(x),x)-a*tan(lambda*x)^n*b*(lambda*tan(lambda* 
x)^2-a*b*tan(lambda*x)^(n+2)+lambda)*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 symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Whittaker 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            -> hypergeometric 
               -> heuristic approach 
               -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebi\ 
us 
            -> Mathieu 
               -> Equivalence to the rational form of Mathieu ODE under a powe\ 
r @ Moebius 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under \ 
a power @ Moebius 
         -> 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 a symmetry of the form [xi=0, eta=F(x)] 
            trying 2nd order exact linear 
            trying symmetries linear in x and y(x) 
            trying to convert to a linear ODE with constant coefficients 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying an equivalence, under non-integer power transformations, 
            to LODEs admitting Liouvillian solutions. 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Whittaker 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            -> hypergeometric 
               -> heuristic approach 
               -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebi\ 
us 
            -> Mathieu 
               -> Equivalence to the rational form of Mathieu ODE under a powe\ 
r @ Moebius 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under \ 
a power @ Moebius 
         -> 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 a symmetry of the form [xi=0, eta=F(x)] 
            trying 2nd order exact linear 
            trying symmetries linear in x and y(x) 
            trying to convert to a linear ODE with constant coefficients 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying 2nd order, integrating factor of the form mu(x,y) 
         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 symmetry of the form [xi=0, eta=F(x)] 
            checking if the LODE is missing y 
            -> Trying a solution in terms of special functions: 
               -> Bessel 
               -> elliptic 
               -> Legendre 
               -> Whittaker 
                  -> hyper3: Equivalence to 1F1 under a power @ Moebius 
               -> hypergeometric 
                  -> heuristic approach 
                  -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Mo\ 
ebius 
               -> Mathieu 
                  -> Equivalence to the rational form of Mathieu ODE under a p\ 
ower @ Moebius 
            -> Heun: Equivalence to the GHE or one of its 4 confluent cases und\ 
er a power @ Moebius 
            -> Heun: Equivalence to the GHE or one of its 4 confluent cases und\ 
er a power @ Moebius 
            -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = e\ 
xp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
               trying a symmetry of the form [xi=0, eta=F(x)] 
               trying 2nd order exact linear 
               trying symmetries linear in x and y(x) 
               trying to convert to a linear ODE with constant coefficients 
            trying a symmetry of the form [xi=0, eta=F(x)] 
            checking if the LODE is missing y 
            -> Trying an equivalence, under non-integer power transformations, 
               to LODEs admitting Liouvillian solutions. 
            -> Trying a solution in terms of special functions: 
               -> Bessel 
               -> elliptic 
               -> Legendre 
               -> Whittaker 
                  -> hyper3: Equivalence to 1F1 under a power @ Moebius 
               -> hypergeometric 
                  -> heuristic approach 
                  -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Mo\ 
ebius 
               -> Mathieu 
                  -> Equivalence to the rational form of Mathieu ODE under a p\ 
ower @ Moebius 
            -> Heun: Equivalence to the GHE or one of its 4 confluent cases und\ 
er a power @ Moebius 
            -> Heun: Equivalence to the GHE or one of its 4 confluent cases und\ 
er a power @ Moebius 
            -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = e\ 
xp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
               trying a symmetry of the form [xi=0, eta=F(x)] 
               trying 2nd order exact linear 
               trying symmetries linear in x and y(x) 
               trying to convert to a linear ODE with constant coefficients 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         trying to convert to an ODE of Bessel type 
         -> trying with_periodic_functions in the coefficients 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE, diff(y(x),x)-(a*tan(lambda*x)^n*y(x)^2+y(x) 
+x^2*(-a*b^2*tan(lambda*x)^(n+2)+b*lambda*tan(lambda*x)^2+b*lambda))/x, y(x), 
explicit 
      *** Sublevel 2 *** 
      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 sub-methods: 
         trying Riccati_symmetries 
      trying inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> 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 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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \tan \left (\lambda x \right )^{13396} y \left (x \right )^{2}-a \,b^{2} \tan \left (\lambda x \right )^{13398}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \tan \left (\lambda x \right )^{13396} y \left (x \right )^{2}-a \,b^{2} \tan \left (\lambda x \right )^{13398}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \end {array} \]

2.11.8.3 ✗ Mathematica

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

Not solved

2.11.8.4 ✗ Sympy

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

Timed Out