60.3.416 problem 1422

Internal problem ID [11426]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1422
Date solved : Tuesday, January 28, 2025 at 06:05:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.443 (sec). Leaf size: 29

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

\[ y = -i \cot \left (x \right ) \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) c_{2} +c_{1} \cot \left (x \right )-2 c_{2} \]

✓ Solution by Mathematica

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

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

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