Internal problem ID [8796]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1218.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (\cot \left (x \right ) x +a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve(x^2*diff(diff(y(x),x),x)+(2*x^2*cot(x)+x)*diff(y(x),x)+(x*cot(x)+a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \csc \left (x \right ) \operatorname {BesselJ}\left (i \sqrt {a}, x\right )+c_{2} \csc \left (x \right ) \operatorname {BesselY}\left (i \sqrt {a}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.092 (sec). Leaf size: 37
DSolve[(a + x*Cot[x])*y[x] + (x + 2*x^2*Cot[x])*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \csc (x) \left (c_1 \operatorname {BesselJ}\left (i \sqrt {a},x\right )+c_2 Y_{i \sqrt {a}}(x)\right ) \\ \end{align*}