60.3.405 problem 1422

Internal problem ID [11401]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1422
Date solved : Sunday, March 30, 2025 at 08:20:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.167 (sec). Leaf size: 29

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

\[ y = -i \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) c_2 \cot \left (x \right )+c_1 \cot \left (x \right )-2 c_2 \]

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 42

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

\[ y(x)\to -\frac {c_2 \cos (x) \arcsin (\cos (x))}{\sqrt {\sin ^2(x)}}+\frac {c_1 \cos (x)}{\sqrt {-\sin ^2(x)}}-c_2 \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve -2*y(x)/sin(x)**2 + Derivative(y(x), (x, 2))