Internal problem ID [9014]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1435.
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 {4 \sin \left (3 x \right ) y}{\sin \relax (x )^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.061 (sec). Leaf size: 41
dsolve(diff(diff(y(x),x),x) = -4*sin(3*x)/sin(x)^3*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sqrt {\sin }\relax (x )\right ) \LegendreP \left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \relax (x )\right )+c_{2} \left (\sqrt {\sin }\relax (x )\right ) \LegendreQ \left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.097 (sec). Leaf size: 61
DSolve[y''[x] == -4*Csc[x]^3*Sin[3*x]*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{-\frac {1}{2}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))+c_2 Q_{-\frac {1}{2}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))\right ) \\ \end{align*}