55.10.7 problem 20

Internal problem ID [13414]
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, October 01, 2025 at 11:01:23 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.071 (sec). Leaf size: 120

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

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

✓ 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*}

✗ 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