Internal problem ID [9008]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1429.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\cos \relax (x ) y^{\prime }}{\sin \relax (x )}-\frac {y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 25
dsolve(diff(diff(y(x),x),x) = -1/sin(x)*cos(x)*diff(y(x),x)+1/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {\sin \relax (x ) c_{1}}{\cos \relax (x )-1}+\frac {\left (\cos \relax (x )-1\right ) c_{2}}{\sin \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 19
DSolve[y''[x] == Csc[x]^2*y[x] - Cot[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \csc (x) (c_1-i c_2 \cos (x)) \\ \end{align*}