55.9.6 problem 6

Internal problem ID [13400]
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 : 6
Date solved : Wednesday, October 01, 2025 at 10:02:17 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 51

ode:=diff(y(x),x) = lambda*sin(lambda*x)*y(x)^2+lambda*sin(lambda*x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \,{\mathrm e}^{\frac {\cos \left (2 \lambda x \right )}{2}+\frac {1}{2}} c_1 -\cos \left (\lambda x \right ) \sqrt {\pi }\, \left (\operatorname {erfi}\left (\cos \left (\lambda x \right )\right ) c_1 +1\right )}{\sqrt {\pi }\, \left (\operatorname {erfi}\left (\cos \left (\lambda x \right )\right ) c_1 +1\right )} \]

✗ Mathematica

ode=D[y[x],x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+\[Lambda]*Sin[\[Lambda]*x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-lambda_*y(x)**2*sin(lambda_*x) - lambda_*sin(lambda_*x)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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