Internal problem ID [9501]
Internal file name [OUTPUT/8441_Monday_June_06_2022_03_05_02_AM_73050089/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1172.
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.
\[ \boxed {x^{2} y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+y a=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type 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: 0.047 (sec). Leaf size: 47
dsolve(x^2*diff(diff(y(x),x),x)+2*(x-1)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {\frac {1}{x}}\, {\mathrm e}^{-\frac {1}{x}} \left (c_{1} \operatorname {BesselI}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right )+c_{2} \operatorname {BesselK}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.157 (sec). Leaf size: 145
DSolve[a*y[x] + 2*(-1 + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (\frac {1}{x}\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (2^{\sqrt {1-4 a}} c_2 \left (\frac {1}{x}\right )^{\sqrt {1-4 a}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (\sqrt {1-4 a}+1\right ),\sqrt {1-4 a}+1,-\frac {2}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a},1-\sqrt {1-4 a},-\frac {2}{x}\right )\right ) \]