Internal problem ID [9765]
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]]
\[ \boxed {y^{\prime \prime }-\frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}+2 y=0} \]
✓ Solution by Maple
Time used: 0.219 (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)
\[ y \left (x \right ) = c_{1} \sin \left (2 x \right )^{\frac {3}{4}} \operatorname {LegendreP}\left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right )+c_{2} \sin \left (2 x \right )^{\frac {3}{4}} \operatorname {LegendreQ}\left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right ) \]
✓ Solution by Mathematica
Time used: 20.33 (sec). Leaf size: 64
DSolve[y''[x] == -2*y[x] + Cot[2*x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {2}{3} c_2 \cos (2 x) \cos ^{\frac {3}{2}}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\cos ^2(x)\right )+\frac {1}{2} c_1 \cos (2 x)-2 c_2 \sin ^2(x)^{3/4} \cos ^{\frac {3}{2}}(x) \]