58.2.20 problem 21

Internal problem ID [9143]
Book : Second order enumerated odes
Section : section 2
Problem number : 21
Date solved : Monday, January 27, 2025 at 05:48:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 23

dsolve(diff(y(x),x$2)+cot(x)*diff(y(x),x)+4*y(x)*csc(x)^2=0,y(x), singsol=all)
 

\[ y = c_{1} \left (\csc \left (x \right )+\cot \left (x \right )\right )^{-2 i}+c_{2} \left (\csc \left (x \right )+\cot \left (x \right )\right )^{2 i} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 25

DSolve[D[y[x],{x,2}]+Cot[x]*D[y[x],x]+4*y[x]*Csc[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cos (2 \text {arctanh}(\cos (x)))-c_2 \sin (2 \text {arctanh}(\cos (x))) \]