2.10.6 Problem 21

Internal problem ID [13383]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with cosine.
Problem number : 21
Date solved : Wednesday, December 31, 2025 at 02:13:33 PM
CAS classification : [_Riccati]

2.10.6.1 Solved using first_order_ode_riccati

41.705 (sec)

Entering first order ode riccati solver

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

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= y^{2} \cos \left (\lambda x \right )^{2} a +\lambda y^{2}+a \cos \left (\lambda x \right )^{2}-a +\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 \(f_0(x)=a \cos \left (\lambda x \right )^{2}-a +\lambda \), \(f_1(x)=0\) and \(f_2(x)=\lambda +a \cos \left (\lambda x \right )^{2}\). Let

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

Substituting the above terms back in equation (2) gives

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

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 ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}+c_2 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -c_1 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}-\frac {c_1 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}}{2}-c_2 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x -\frac {c_2 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x}{2}+2 i c_2 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right ) \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \left (\lambda +a \cos \left (\lambda x \right )^{2}\right )} \\ y &= -\frac {-c_1 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}-\frac {c_1 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}}{2}-c_2 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x -\frac {c_2 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x}{2}+2 i c_2 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )}{\left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) \left (c_1 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}+c_2 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x \right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {-\lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}-\frac {\cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}}{2}-c_3 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x -\frac {c_3 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x}{2}+2 i c_3 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )}{\left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) \left (\cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}+c_3 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x \right )} \]

Summary of solutions found

\begin{align*} y &= -\frac {-\lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}-\frac {\cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}}{2}-c_3 \lambda \sin \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x -\frac {c_3 \cos \left (\lambda x \right ) \sin \left (2 \lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x}{2}+2 i c_3 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )}{\left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) \left (\cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }}+c_3 \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{4 \lambda }} \int 2 i \lambda \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x \right )} \\ \end{align*}

2.10.6.2 ✓ Maple. Time used: 0.005 (sec). Leaf size: 102

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

\[ y = \frac {-2 \lambda \int {\mathrm e}^{-\frac {\cos \left (2 \lambda x \right ) a}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x c_1 \tan \left (\lambda x \right )+i \tan \left (\lambda x \right )+2 \,{\mathrm e}^{-\frac {\cos \left (2 \lambda x \right ) a}{2 \lambda }} c_1 \lambda \sec \left (\lambda x \right )^{2}}{-2 \lambda \int {\mathrm e}^{-\frac {\cos \left (2 \lambda x \right ) a}{2 \lambda }} \left (\lambda \sec \left (\lambda x \right )^{2}+a \right )d x c_1 +i} \]

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) = -2*a*cos(lambda*x)* 
lambda*sin(lambda*x)/(lambda+a*cos(lambda*x)^2)*diff(y(x),x)-(lambda+a*cos( 
lambda*x)^2)*(a*cos(lambda*x)^2-a+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 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 
               -> Kummer 
                  -> 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 \ 
under a power @ Moebius 
            No special function solution was found. 
         <- Kovacics algorithm successful 
         Change of variables used: 
            [x = 1/2*arccos(t)/lambda] 
         Linear ODE actually solved: 
            (4*a^3*t^3+4*a^3*t^2+24*a^2*lambda*t^2-4*a^3*t+16*a^2*lambda*t+48*a\ 
*lambda^2*t-4*a^3-8*a^2*lambda+16*a*lambda^2+32*lambda^3)*u(t)+(-64*a*lambda^2*\ 
t-128*lambda^3*t-64*a*lambda^2)*diff(u(t),t)+(-64*a*lambda^2*t^3-64*a*lambda^2*\ 
t^2-128*lambda^3*t^2+64*a*lambda^2*t+64*a*lambda^2+128*lambda^3)*diff(diff(u(t)\ 
,t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.10.6.3 ✓ Mathematica. Time used: 7.995 (sec). Leaf size: 191

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

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

2.10.6.4 ✗ Sympy

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

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