Internal problem ID [10969]
Internal file name [OUTPUT/10226_Sunday_December_31_2023_11_10_23_AM_15916150/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: 146.
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.
\[ \boxed {x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y=0} \]
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 <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 148
dsolve(x^2*diff(y(x),x$2)+x*(a*x^n+b)*diff(y(x),x)+(alpha*x^(2*n)+beta*x^n+gamma)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{\frac {1}{2}-\frac {b}{2}-\frac {n}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a \left (b +n -1\right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, n}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha }\, x^{n}}{n}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a \left (b +n -1\right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, n}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha }\, x^{n}}{n}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.485 (sec). Leaf size: 420
DSolve[x^2*y''[x]+x*(a*x^n+b)*y'[x]+(\[Alpha]*x^(2*n)+\[Beta]*x^n+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{\frac {1}{2}-\frac {n}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2}+1\right )} e^{-\frac {\left (\sqrt {a^2-4 \alpha }+a\right ) x^n}{2 n}} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}-b n+n^2}{2 n^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right ) a^2+n (b+n-1) \sqrt {a^2-4 \alpha } a-2 \left (2 \alpha n^2+\sqrt {a^2-4 \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 \alpha }}{n}\right )+c_2 L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right ) a^2\right )-n (b+n-1) \sqrt {a^2-4 \alpha } a+4 n^2 \alpha +2 n \sqrt {a^2-4 \alpha } \beta +4 \alpha \sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 n^2 \left (a^2-4 \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 \alpha }}{n}\right )\right ) \]