Internal problem ID [9010]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1431.
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 {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}+2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 35
dsolve(diff(diff(y(x),x),x) = cos(2*x)/sin(2*x)*diff(y(x),x)-2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sin ^{\frac {3}{4}}\left (2 x \right )\right ) \LegendreP \left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right )+c_{2} \left (\sin ^{\frac {3}{4}}\left (2 x \right )\right ) \LegendreQ \left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right ) \]
✓ Solution by Mathematica
Time used: 10.109 (sec). Leaf size: 58
DSolve[y''[x] == -2*y[x] + Cot[2*x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} c_1 \cos (2 x)-\frac {2}{3} c_2 \cos ^{\frac {3}{2}}(x) \left (\cos (2 x) \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\cos ^2(x)\right )+3 \sin ^2(x)^{3/4}\right ) \\ \end{align*}