2.13.8 Problem 57

Internal problem ID [13416]
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 : 57
Date solved : Sunday, January 18, 2026 at 08:02:21 PM
CAS classification : [_Riccati]

2.13.8.1 Solved using first_order_ode_riccati

9.984 (sec)

Entering first order ode riccati solver

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

In canonical form the ODE is

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

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = -\tan \left (\lambda x \right )^{2} a^{2}+\tan \left (\lambda x \right )^{2} a \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 a b +a \lambda +b \lambda +y^{2} \]

With Riccati ODE standard form

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

Shows that \(f_0(x)=-\tan \left (\lambda x \right )^{2} a^{2}+\tan \left (\lambda x \right )^{2} a \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 a b +a \lambda +b \lambda \), \(f_1(x)=0\) and \(f_2(x)=1\). Let

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

Substituting the above terms back in equation (2) gives

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

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 ) = c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }}+c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -\frac {c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} a \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {b}{\lambda }}}{\cos \left (\lambda x \right )}+\frac {c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }} b \cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}-\frac {c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \left (\lambda -a \right ) \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\cos \left (\lambda x \right )}+\frac {c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (\lambda -b \right ) \cos \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\sin \left (\lambda x \right )}+\frac {4 c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (-b +\lambda -a \right ) \operatorname {hypergeom}\left (\left [2, \frac {-b +\lambda -a}{\lambda }+1\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }+1\right ], \cos \left (\lambda x \right )^{2}\right ) \cos \left (\lambda x \right ) \lambda \sin \left (\lambda x \right )}{-3 \lambda +2 a} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u} \\ y &= -\frac {-\frac {c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} a \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {b}{\lambda }}}{\cos \left (\lambda x \right )}+\frac {c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }} b \cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}-\frac {c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \left (\lambda -a \right ) \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\cos \left (\lambda x \right )}+\frac {c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (\lambda -b \right ) \cos \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\sin \left (\lambda x \right )}+\frac {4 c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (-b +\lambda -a \right ) \operatorname {hypergeom}\left (\left [2, \frac {-b +\lambda -a}{\lambda }+1\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }+1\right ], \cos \left (\lambda x \right )^{2}\right ) \cos \left (\lambda x \right ) \lambda \sin \left (\lambda x \right )}{-3 \lambda +2 a}}{c_1 \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }}+c_2 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {-\frac {\cos \left (\lambda x \right )^{\frac {a}{\lambda }} a \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {b}{\lambda }}}{\cos \left (\lambda x \right )}+\frac {\cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }} b \cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}-\frac {c_3 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \left (\lambda -a \right ) \sin \left (\lambda x \right ) \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\cos \left (\lambda x \right )}+\frac {c_3 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (\lambda -b \right ) \cos \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}{\sin \left (\lambda x \right )}+\frac {4 c_3 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \left (-b +\lambda -a \right ) \operatorname {hypergeom}\left (\left [2, \frac {-b +\lambda -a}{\lambda }+1\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }+1\right ], \cos \left (\lambda x \right )^{2}\right ) \cos \left (\lambda x \right ) \lambda \sin \left (\lambda x \right )}{-3 \lambda +2 a}}{\cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }}+c_3 \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {4 c_3 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a -\lambda +b \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 \left (\left (-3 \lambda ^{2}+\frac {\left (7 a +3 b \right ) \lambda }{2}-a b \right ) \cos \left (\lambda x \right )^{2}+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \lambda \left (a -\frac {3 \lambda }{5}\right )}{2}\right ) c_3 \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \left (-\frac {3 \lambda }{2}+a \right ) \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (\lambda x \right )-\cot \left (\lambda x \right ) b \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (\cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+c_3 \cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {4 c_3 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a -\lambda +b \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 \left (\left (-3 \lambda ^{2}+\frac {\left (7 a +3 b \right ) \lambda }{2}-a b \right ) \cos \left (\lambda x \right )^{2}+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \lambda \left (a -\frac {3 \lambda }{5}\right )}{2}\right ) c_3 \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \left (-\frac {3 \lambda }{2}+a \right ) \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (\lambda x \right )-\cot \left (\lambda x \right ) b \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (\cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+c_3 \cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \\ \end{align*}

2.13.8.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 268

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

\[ y = \frac {4 c_1 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (\lambda x \right )^{2}+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \lambda \left (a -\frac {3 \lambda }{5}\right )}{2}\right ) c_1 \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \left (a -\frac {3 \lambda }{2}\right ) \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_1 \cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}\right )} \]

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*tan(lambda*x)^2 
-a*tan(lambda*x)^2*lambda+b^2*cot(lambda*x)^2-b*cot(lambda*x)^2*lambda-2*a*b- 
lambda*a-lambda*b)*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 
               -> Whittaker 
                  -> hyper3: Equivalence to 1F1 under a power @ Moebius 
               -> hypergeometric 
                  -> heuristic approach 
                  <- heuristic approach successful 
               <- hypergeometric successful 
            <- special function solution successful 
               -> Trying to convert hypergeometric functions to elementary form\ 
... 
               <- elementary form for at least one hypergeometric solution is a\ 
chieved - returning with no uncomputed integrals 
            <- Kovacics algorithm successful 
         Change of variables used: 
            [x = 1/lambda*arccos(t)] 
         Linear ODE actually solved: 
            (a^2*t^4+2*a*b*t^4+b^2*t^4-2*a^2*t^2-2*a*b*t^2+a*lambda*t^2-b*lambd\ 
a*t^2+a^2-a*lambda)*u(t)+(-lambda^2*t^5+lambda^2*t^3)*diff(u(t),t)+(-lambda^2*t\ 
^6+2*lambda^2*t^4-lambda^2*t^2)*diff(diff(u(t),t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.13.8.3 ✗ Mathematica

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

Not solved

2.13.8.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-2*a*b - a*lambda_ - a*(-a + lambda_)*tan(lambda_*x)**2 - b*lambda_ - b*(-b + lambda_)/tan(lambda_*x)**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 - 2*a*b - a*lambda_*tan(lambda_*x)**2 - a
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', 'lie_group')