Internal problem ID [11077]
Internal file name [OUTPUT/10334_Wednesday_January_24_2024_10_16_50_PM_90181001/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: 254.
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.
\[ \boxed {x^{2} \left (x^{2 n} a^{2}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y=0} \]
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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunG ODE, case a <> 0, e <> 0, g <> 0, c = 0 `
✓ Solution by Maple
Time used: 1.734 (sec). Leaf size: 273
dsolve(x^2*(a^2*x^(2*n)-1)*diff(y(x),x$2)+x*(a*p*x^n+q)*diff(y(x),x)+(a*r*x^n+s)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{\frac {q}{2}+\frac {1}{2}} \left (c_{1} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {HeunG}\left (-1, \frac {p q +\sqrt {q^{2}+2 q +4 s +1}\, p +p +2 r}{2 n^{2}}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q -1}{2 n}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q +1}{2 n}, \frac {n +\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )+c_{2} x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {HeunG}\left (-1, \frac {-\sqrt {q^{2}+2 q +4 s +1}\, p +\left (q +1\right ) p +2 r}{2 n^{2}}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q -1}{2 n}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q +1}{2 n}, \frac {n -\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*(a^2*x^(2*n)-1)*y''[x]+x*(a*p*x^n+q)*y'[x]+(a*r*x^n+s)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved