Internal problem ID [10964]
Internal file name [OUTPUT/10221_Sunday_December_31_2023_11_10_17_AM_90655406/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 141.
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.
\[ \boxed {x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y=0} \]
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: 0.125 (sec). Leaf size: 81
dsolve(x^2*diff(y(x),x$2)-2*x*(x^2-a)*diff(y(x),x)+(2*n*x^2+( (-1)^n-1)*a )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-a -\frac {1}{2}} {\mathrm e}^{\frac {x^{2}}{2}} \left (\operatorname {WhittakerM}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{1} +\operatorname {WhittakerW}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.646 (sec). Leaf size: 231
DSolve[x^2*y''[x]-2*x*(x^2-a)*y'[x]+(2*n*x^2+( (-1)^n-1)*a )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to i^{-a} (-1)^{\frac {1}{4} \left (1-\sqrt {4 a^2-4 a (-1)^n+1}\right )} x^{\frac {1}{2} \left (-\sqrt {4 a^2-4 a (-1)^n+1}-2 a+1\right )} \left (c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n-\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),1-\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1},x^2\right )+c_2 i^{\sqrt {4 a^2-4 a (-1)^n+1}} x^{\sqrt {4 a^2-4 a (-1)^n+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )\right ) \]