Internal problem ID [8996]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1417.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (\sin ^{2}\relax (x )-\cos \relax (x )\right ) y^{\prime }}{\sin \relax (x )}+y \left (\sin ^{2}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 35
dsolve(diff(diff(y(x),x),x) = -(sin(x)^2-cos(x))/sin(x)*diff(y(x),x)-y(x)*sin(x)^2,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {\cos \relax (x )}{2}} \sin \left (\frac {\sqrt {3}\, \cos \relax (x )}{2}\right )+c_{2} {\mathrm e}^{\frac {\cos \relax (x )}{2}} \cos \left (\frac {\sqrt {3}\, \cos \relax (x )}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 45
DSolve[y''[x] == -(Sin[x]^2*y[x]) - Csc[x]*(-Cos[x] + Sin[x]^2)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {\cos (x)}{2}} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \cos (x)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \cos (x)\right )\right ) \\ \end{align*}