Internal problem ID [9547]
Internal file name [OUTPUT/8487_Monday_June_06_2022_03_12_38_AM_96366425/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 28
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)
\[ y \left (x \right ) = \csc \left (x \right ) \left (c_{1} \operatorname {BesselJ}\left (i \sqrt {a}, x\right )+c_{2} \operatorname {BesselY}\left (i \sqrt {a}, x\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.174 (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]
\[ y(x)\to \csc (x) \left (c_1 \operatorname {BesselJ}\left (i \sqrt {a},x\right )+c_2 \operatorname {BesselY}\left (i \sqrt {a},x\right )\right ) \]