Problem 15

2.10.2 Problem 15

2.10.2.1 Solved using ﬁrst_order_ode_riccati
2.10.2.2 ✓Maple
2.10.2.3 ✓Mathematica
2.10.2.4 ✗Sympy

Internal problem ID [13379]
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 : 15
Date solved : Wednesday, December 31, 2025 at 02:09:56 PM
CAS classiﬁcation : [_Riccati]

2.10.2.1 Solved using ﬁrst_order_ode_riccati

17.312 (sec)

Entering ﬁrst order ode riccati solver

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

In canonical form the ODE is

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

Substituting the above terms back in equation (2) gives

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

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 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -c_1 \sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\frac {c_1 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}-c_2 \sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )-\frac {c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )}{2}-\frac {c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\lambda x}{2}\right ) \lambda }{2} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u} \\ y &= -\frac {-c_1 \sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\frac {c_1 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}-c_2 \sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )-\frac {c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )}{2}-\frac {c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\lambda x}{2}\right ) \lambda }{2}}{c_1 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {-\sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\frac {{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}-c_3 \sin \left (\lambda x \right ) a \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )-\frac {c_3 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \sin \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\lambda x}{2}\right ) \lambda }{2}}{{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_3 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {\sin \left (\frac {\lambda x}{2}\right ) \left (c_3 \left (\cos \left (\lambda x \right ) a +a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\left (\cos \left (\lambda x \right )+1\right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_3 \lambda }{2}+2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) a \cos \left (\frac {\lambda x}{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right ) \lambda \right )}{\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_3 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \\ \end{align*}

Summary of solutions found

2.10.2.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 272

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

\[ y = \frac {\left (2 a c_1 \sin \left (\lambda x \right ) \cos \left (\frac {\lambda x}{2}\right )+c_1 \lambda \sin \left (\frac {\lambda x}{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+2 \sin \left (\lambda x \right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_1 \cos \left (\frac {\lambda x}{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{2}\right )}{2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right ) c_1 +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \]

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-a*lambda*cos( 
lambda*x)-a^2*cos(lambda*x)^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 
               -> 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 
               <- Heun successful: received ODE is equivalent to the  HeunC  OD\ 
E, case  a <> 0, e <> 0, c = 0 
            <- Kovacics algorithm successful 
         Change of variables used: 
            [x = arccos(t)/lambda] 
         Linear ODE actually solved: 
            (2*a^2*t^2+2*a*lambda*t-2*a^2)*u(t)-2*lambda^2*t*diff(u(t),t)+(-2*l\ 
ambda^2*t^2+2*lambda^2)*diff(diff(u(t),t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful

Maple step by step

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

2.10.2.3 ✓ Mathematica. Time used: 1.167 (sec). Leaf size: 156

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

\begin{align*} y(x)&\to \frac {c_1 \left (-\exp \left (\int _1^x2 a \sin (\lambda K[1])dK[1]\right )\right )+a c_1 \sin (\lambda x) \int _1^x\exp \left (\int _1^{K[2]}2 a \sin (\lambda K[1])dK[1]\right )dK[2]+a \sin (\lambda x)}{1+c_1 \int _1^x\exp \left (\int _1^{K[2]}2 a \sin (\lambda K[1])dK[1]\right )dK[2]}\\ y(x)&\to a \sin (\lambda x)-\frac {\exp \left (\int _1^x2 a \sin (\lambda K[1])dK[1]\right )}{\int _1^x\exp \left (\int _1^{K[2]}2 a \sin (\lambda K[1])dK[1]\right )dK[2]} \end{align*}

2.10.2.4 ✗ Sympy

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

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

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