2.10.5 Problem 20

Internal problem ID [13382]
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 : 20
Date solved : Wednesday, December 31, 2025 at 02:12:04 PM
CAS classification : [_Riccati]

2.10.5.1 Solved using first_order_ode_riccati

36.366 (sec)

Entering first order ode riccati solver

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

In canonical form the ODE is

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

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

Substituting the above terms back in equation (2) gives

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

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 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+c_2 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_1 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_2 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+\frac {c_2 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+i c_2 \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \left (-\frac {\cos \left (\lambda x \right ) a}{2}+\frac {a}{2}+\frac {\lambda }{2}\right )} \\ y &= -\frac {\frac {c_1 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_1 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_2 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+\frac {c_2 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+i c_2 \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\left (-\frac {\cos \left (\lambda x \right ) a}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) \left (c_1 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+c_2 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x \right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {\lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {\sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_3 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+\frac {c_3 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+i c_3 \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\left (-\frac {\cos \left (\lambda x \right ) a}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) \left (\sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+c_3 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x \right )} \]

Summary of solutions found

\begin{align*} y &= -\frac {\frac {\lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {\sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{2}+\frac {c_3 \lambda \cos \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+\frac {c_3 \sin \left (\frac {\lambda x}{2}\right ) \sin \left (\lambda x \right ) a \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x}{2}+i c_3 \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\left (-\frac {\cos \left (\lambda x \right ) a}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) \left (\sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+c_3 \sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \int \frac {i \left (\frac {\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}}{2}+a \right ) \sqrt {\frac {1}{2}-\frac {\cos \left (\lambda x \right )}{2}}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda }{\sin \left (\frac {\lambda x}{2}\right )}d x \right )} \\ \end{align*}

2.10.5.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 122

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

\[ y = \frac {-2 \,\operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} c_1 \lambda \csc \left (\frac {\lambda x}{2}\right )^{2}-\lambda \int {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \left (\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}+2 a \right ) \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right )d x c_1 \cot \left (\frac {\lambda x}{2}\right )+2 i \cot \left (\frac {\lambda x}{2}\right )}{\lambda \int {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \left (\lambda \csc \left (\frac {\lambda x}{2}\right )^{2}+2 a \right ) \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right )d x c_1 -2 i} \]

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) = -a*lambda*sin(lambda 
*x)/(-lambda-a+a*cos(lambda*x))*diff(y(x),x)-1/4*(-lambda-a+a*cos(lambda*x))*(a 
*cos(lambda*x)+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 = arccos(t)/lambda] 
         Linear ODE actually solved: 
            (4*a^3*t^3-4*a^3*t^2-12*a^2*lambda*t^2-4*a^3*t+8*a^2*lambda*t+12*a*\ 
lambda^2*t+4*a^3+4*a^2*lambda-4*a*lambda^2-4*lambda^3)*u(t)+(16*a*lambda^2*t+16\ 
*lambda^3*t-16*a*lambda^2)*diff(u(t),t)+(-16*a*lambda^2*t^3+16*a*lambda^2*t^2+1\ 
6*lambda^3*t^2+16*a*lambda^2*t-16*a*lambda^2-16*lambda^3)*diff(diff(u(t),t),t) \ 
= 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.10.5.3 ✓ Mathematica. Time used: 7.324 (sec). Leaf size: 234

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

\begin{align*} y(x)&\to -\frac {2 \left (c_1 \cot \left (\frac {\lambda x}{2}\right ) \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+2 c_1 \csc ^2\left (\frac {\lambda x}{2}\right ) e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\cot \left (\frac {\lambda x}{2}\right )\right )}{2+2 c_1 \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}\\ y(x)&\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \end{align*}

2.10.5.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a*cos(lambda_*x) + a - lambda_ - (-a*cos(lambda_*x) + a + lambda_)*y(x)**2 + 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*y(x)**2/2 + a*cos(lambda_*x)/2 + a/2 - lambda_*y(x)**2/2 - lambda_/2 + Derivative(y(x), x) cannot be solved by the factorable group method