61.9.7 problem 7

Internal problem ID [12102]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with sine
Problem number : 7
Date solved : Sunday, March 30, 2025 at 10:44:42 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 220

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

\[ y = -\frac {\left (\left (a \cos \left (\lambda x \right )+\tan \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (\left (\textit {\_a} -1\right ) a -\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{{3}/{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} c_1 +1\right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )+\frac {{\mathrm e}^{\frac {a \sin \left (\lambda x \right )}{\lambda }} \cos \left (\lambda x \right ) c_1 \sec \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2} \csc \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \left (-\lambda -a +a \sin \left (\lambda x \right )\right )}{2}\right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )}{\sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \left (-\lambda -a +a \sin \left (\lambda x \right )\right ) \left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (\left (\textit {\_a} -1\right ) a -\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{{3}/{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} c_1 +1\right )} \]

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a*sin(lambda_*x) + a - lambda_ - (-a*sin(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*sin(lambda_*x)/2 - a*y(x)**2/2 + a*sin(lambda_*x)/2 + a/2 - lambda_*y(x)**2/2 - lambda_/2 + Derivative(y(x), x) cannot be solved by the factorable group method