61.10.6 problem 19

Internal problem ID [12114]
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 : 19
Date solved : Sunday, March 30, 2025 at 10:54:06 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 95

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

\[ y = -\frac {4 \left (-\frac {\sin \left (\lambda x \right )^{2} \left (c_1 -\frac {1}{2}\right ) \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )}{2}+\left (c_1 -\frac {1}{2}\right ) {\mathrm e}^{\sin \left (\lambda x \right )^{2}} \sqrt {-\sin \left (\lambda x \right )^{2}}+\frac {\sqrt {\pi }\, \sin \left (\lambda x \right )^{2} c_1}{2}\right ) \csc \left (\lambda x \right )}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right ) \left (2 c_1 -1\right )-2 c_1 \right )} \]

✗ Mathematica

ode=D[y[x],x]==\[Lambda]*Cos[\[Lambda]*x]*y[x]^2+\[Lambda]*Cos[\[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*cos(lambda_*x) - lambda_*cos(lambda_*x)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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