2.13.2 Problem 50

Internal problem ID [13410]
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 : 50
Date solved : Friday, December 19, 2025 at 04:02:58 AM
CAS classification : [_Riccati]

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

Entering first order ode riccati solver

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

In canonical form the ODE is

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

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \cos \left (\lambda x \right ) a \,y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n} \]

With Riccati ODE standard form

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

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\cos \left (\lambda x \right ) a 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' &=-a \lambda \sin \left (\lambda x \right )\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\cos \left (\lambda x \right )^{3} a^{2} b \sin \left (\lambda x \right )^{n} \end{align*}

Substituting the above terms back in equation (2) gives

\[ \cos \left (\lambda x \right ) a u^{\prime \prime }\left (x \right )+a \lambda \sin \left (\lambda x \right ) u^{\prime }\left (x \right )+\cos \left (\lambda x \right )^{3} a^{2} b \sin \left (\lambda x \right )^{n} u \left (x \right ) = 0 \]

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

Taking derivative gives

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

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{\cos \left (\lambda x \right ) a u} \\ y &= \text {Expression too large to display} \\ \end{align*}

Doing change of constants, the above solution becomes

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

Simplifying the above gives

\begin{align*} \text {Expression too large to display} \\ \end{align*}

The solution

\[ \text {Expression too large to display} \]

was found not to satisfy the ode or the IC. Hence it is removed.

2.13.2.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 1039

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

\[ \text {Expression too large to display} \]

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) = -lambda*sin(lambda*x 
)/cos(lambda*x)*diff(y(x),x)-a*cos(lambda*x)^2*b*sin(lambda*x)^n*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 an equivalence, under non-integer power transformations, 
            to LODEs admitting Liouvillian solutions. 
            -> Trying a Liouvillian solution using Kovacics algorithm 
            <- No Liouvillian solutions exists 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Kummer 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
               <- hyper3 successful: received ODE is equivalent to the 0F1 ODE 
            <- Kummer successful 
         <- special function solution successful 
         Change of variables used: 
            [x = arccos(t)/lambda] 
         Linear ODE actually solved: 
            4*a*b*(-t^2+1)^(1/2*n)*t^3*u(t)-4*lambda^2*diff(u(t),t)+(-4*lambda^\ 
2*t^3+4*lambda^2*t)*diff(diff(u(t),t),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

2.13.2.2 ✓ Mathematica. Time used: 0.426 (sec). Leaf size: 633

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

\begin{align*} \text {Solution too large to show}\end{align*}

2.13.2.3 ✗ 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*y(x)**2*cos(lambda_*x) - b*sin(lambda_*x)**n*cos(lambda_*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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