Internal problem ID [9632]
Internal file name [OUTPUT/8574_Monday_June_06_2022_04_13_40_AM_24311131/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1305.
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.
\[ \boxed {x^{3} y^{\prime \prime }+2 y^{\prime } x -y=0} \]
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: 0.047 (sec). Leaf size: 46
dsolve(x^3*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {1}{x}} \left (\operatorname {BesselK}\left (0, -\frac {1}{x}\right ) c_{2} -\operatorname {BesselK}\left (1, -\frac {1}{x}\right ) c_{2} +c_{1} \left (\operatorname {BesselI}\left (0, -\frac {1}{x}\right )+\operatorname {BesselI}\left (1, -\frac {1}{x}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.219 (sec). Leaf size: 47
DSolve[-y[x] + 2*x*y'[x] + x^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {2}{x}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )+c_1 e^{\frac {1}{x}} \left (\operatorname {BesselI}\left (0,\frac {1}{x}\right )-\operatorname {BesselI}\left (1,\frac {1}{x}\right )\right ) \]