Internal problem ID [11076]
Internal file name [OUTPUT/10333_Wednesday_January_24_2024_10_16_49_PM_16279652/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 2.1.2-8. Other equations.
Problem number: 253.
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.
\[ \boxed {x^{2} \left (x^{2 n} a^{2}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (1+n \right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Legendre successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 23
dsolve(x^2*(a^2*x^(2*n)-1)*diff(y(x),x$2)+x*(a^2*(n+1)*x^(2*n)+n-1)*diff(y(x),x)-nu*(nu+1)*a^2*n^2*x^(2*n)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\nu , a \,x^{n}\right )+c_{2} \operatorname {LegendreQ}\left (\nu , a \,x^{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.21 (sec). Leaf size: 79
DSolve[x^2*(a^2*x^(2*n)-1)*y''[x]+x*(a^2*(n+1)*x^(2*n)+n-1)*y'[x]-\[Nu]*(\[Nu]+1)*a^2*n^2*x^(2*n)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to i a c_2 \sqrt {x^{2 n}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {\nu }{2},\frac {\nu }{2}+1,\frac {3}{2},a^2 x^{2 n}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\nu }{2},\frac {\nu +1}{2},\frac {1}{2},a^2 x^{2 n}\right ) \]