Internal problem ID [9676]
Internal file name [OUTPUT/8618_Monday_June_06_2022_04_29_00_AM_55867019/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1349.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}=0} \]
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 -> elliptic -> Legendre <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 65
dsolve(diff(diff(y(x),x),x) = -(x^2+1)/x^3*diff(y(x),x)-1/x^4*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (c_{1} \left (2 x^{2}-1\right ) \operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right )+\left (2 x^{2}-1\right ) c_{2} \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right )+\operatorname {BesselI}\left (1, \frac {1}{4 x^{2}}\right ) c_{1} +\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right ) c_{2} \right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.22 (sec). Leaf size: 73
DSolve[y''[x] == -(y[x]/x^4) - ((1 + x^2)*y'[x])/x^3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| \begin {array}{c} \frac {3}{2} \\ 0,0 \\ \end {array} \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) \operatorname {BesselI}\left (0,\frac {1}{4 x^2}\right )+\operatorname {BesselI}\left (1,\frac {1}{4 x^2}\right )\right )}{2 x^2} \]