50.4.72 problem 69

Internal problem ID [10247]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 69
Date solved : Tuesday, September 30, 2025 at 07:13:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime } \sin \left (2 x \right )^{2}+y^{\prime } \sin \left (4 x \right )-4 y&=0 \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)*sin(2*x)^2+diff(y(x),x)*sin(4*x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \csc \left (2 x \right )+c_2 \cot \left (2 x \right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 29

ode=D[y[x],{x,2}]*Sin[2*x]^2+D[y[x],x]*Sin[4*x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1-i c_2 \cos (2 x)}{\sqrt {\sin ^2(2 x)}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) + sin(2*x)**2*Derivative(y(x), (x, 2)) + sin(4*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(4*y(x) - sin(2*x)**2*Derivative(y(x), (x, 2)))/sin(4*x) + Derivative(y(x), x) cannot be solved by the factorable group method