55.10.8 problem 21

Internal problem ID [13415]
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, October 01, 2025 at 11:02:02 AM
CAS classification : [_Riccati]

\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*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 100

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);
 

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

✓ 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*}

✗ 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