Internal problem ID [10970]
Internal file name [OUTPUT/10227_Sunday_December_31_2023_11_10_24_AM_96377248/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: 147.
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.
\[ \boxed {x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y=0} \]
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: 0.172 (sec). Leaf size: 115
dsolve(x^2*diff(y(x),x$2)+x*(2*a*x^n+b)*diff(y(x),x)+(a^2*x^(2*n)+a*(b+n-1)*x^n+alpha*x^(2*m)+beta*x^m+gamma)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {b}{2}} x^{-\frac {m}{2}} \sqrt {x}\, {\mathrm e}^{-\frac {a \,x^{n}}{n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.47 (sec). Leaf size: 291
DSolve[x^2*y''[x]+x*(2*a*x^n+b)*y'[x]+(a^2*x^(2*n)+a*(b+n-1)*x^n+\[Alpha]*x^(2*m)+\[Beta]*x^m+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{\frac {1}{2}-\frac {m}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} \left (x^n\right )^{-\frac {b}{2 n}} \left (x^m\right )^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} e^{-\frac {a x^n}{n}+\frac {i \sqrt {\alpha } x^m}{m}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2},\frac {m^2+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2},-\frac {2 i x^m \sqrt {\alpha }}{m}\right )+c_2 L_{-\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2}}^{\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}}\left (-\frac {2 i x^m \sqrt {\alpha }}{m}\right )\right ) \]