Internal problem ID [11309]
Internal file name [OUTPUT/10295_Wednesday_December_21_2022_03_47_59_PM_73161115/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VIII, Linear differential equations of the second order. Article 55. Summary.
Page 129
Problem number: Ex 9.
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.
\[ \boxed {x^{2} y^{\prime \prime }-2 n x \left (1+x \right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y=0} \]
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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 89
dsolve(x^2*diff(y(x),x$2)-2*n*x*(1+x)*diff(y(x),x)+(n^2+n+a^2*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{x n} x^{n} \left (\operatorname {WhittakerM}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) c_{1} +\operatorname {WhittakerW}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) c_{2} \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*y''[x]-2*n*x*(1+x)*y'[x]+(n^2+n+a^2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved