2.10.8 Problem 23

Internal problem ID [13385]
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 : 23
Date solved : Wednesday, December 31, 2025 at 02:15:05 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

2.10.8.1 Solved using first_order_ode_riccati

179.064 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \cos \left (\lambda x +\mu \right )^{k} x^{2 n} a \,b^{2}+2 \cos \left (\lambda x +\mu \right )^{k} x^{n} a b c -2 \cos \left (\lambda x +\mu \right )^{k} x^{n} a b y+\cos \left (\lambda x +\mu \right )^{k} a \,c^{2}-2 \cos \left (\lambda x +\mu \right )^{k} a c y+\cos \left (\lambda x +\mu \right )^{k} a y^{2}+b n \,x^{n -1} \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 \(f_0(x)=\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{2 n} a \,b^{2}+2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{n} a b c +\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a \,c^{2}+\frac {b \,x^{n} n}{x}\), \(f_1(x)=-2 x^{n} \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a b -2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} c a\) and \(f_2(x)=\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a\). Let

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

Substituting the above terms back in equation (2) gives

\[ \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a u^{\prime \prime }\left (x \right )-\left (\frac {\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} k \left (-\lambda \sin \left (\lambda x \right ) \cos \left (\mu \right )-\lambda \cos \left (\lambda x \right ) \sin \left (\mu \right )\right ) a}{\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )}+\left (-2 x^{n} \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a b -2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} c a \right ) \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a \right ) u^{\prime }\left (x \right )+\cos \left (\lambda x +\mu \right )^{2 k} a^{2} \left (\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{2 n} a \,b^{2}+2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{n} a b c +\left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a \,c^{2}+\frac {b \,x^{n} n}{x}\right ) u \left (x \right ) = 0 \]

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 \sqrt {\sin \left (\lambda x +\mu \right )}\, \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{-\frac {k}{2}} {\mathrm e}^{-\frac {\int \left (2 a \sec \left (\lambda x +\mu \right ) \left (b \,x^{n}+c \right ) \cos \left (\lambda x +\mu \right )^{k +1}+\lambda \left (\tan \left (\lambda x +\mu \right ) k +\cot \left (\lambda x +\mu \right )\right )\right )d x}{2}}+c_2 \sqrt {\sin \left (\lambda x +\mu \right )}\, \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{-\frac {k}{2}} \int -i \lambda \cos \left (\lambda x +\mu \right )^{k}d x {\mathrm e}^{-\frac {\int \left (2 a \sec \left (\lambda x +\mu \right ) \left (b \,x^{n}+c \right ) \cos \left (\lambda x +\mu \right )^{k +1}+\lambda \left (\tan \left (\lambda x +\mu \right ) k +\cot \left (\lambda x +\mu \right )\right )\right )d x}{2}} \end{equation}

Taking derivative gives

\begin{equation} \tag{4} \text {Expression too large to display} \end{equation}

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

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

Doing change of constants, the above solution becomes

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

Summary of solutions found

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

2.10.8.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 38

ode:=diff(y(x),x) = a*cos(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = b \,x^{n}+c +\frac {1}{c_1 -a \int \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k}d x} \]

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: 
   <- Riccati particular case Kamke (d) successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \cos \left (\lambda x +\mu \right )^{k} \left (y \left (x \right )-b \,x^{13385}-c \right )^{2}+13385 b \,x^{13384} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \cos \left (\lambda x +\mu \right )^{k} \left (y \left (x \right )-b \,x^{13385}-c \right )^{2}+13385 b \,x^{13384} \end {array} \]

2.10.8.3 ✓ Mathematica. Time used: 0.713 (sec). Leaf size: 92

ode=D[y[x],x]==a*Cos[\[Lambda]*x+\[Mu]]^k*(y[x]-b*x^n-c)^2+b*n*x^(n-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{\frac {a \sqrt {\sin ^2(\mu +\lambda x)} \csc (\mu +\lambda x) \cos ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(x \lambda +\mu )\right )}{(k+1) \lambda }+c_1}+b x^n+c\\ y(x)&\to b x^n+c \end{align*}

2.10.8.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
lambda_ = symbols("lambda_") 
mu = symbols("mu") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*(-b*x**n - c + y(x))**2*cos(lambda_*x + mu)**k - b*n*x**(n - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out