Internal problem ID [10870]
Internal file name [OUTPUT/10127_Sunday_December_24_2023_05_13_44_PM_32460967/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 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 47.
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.
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y=0} \]
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 -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 171
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)+(b*x^(2*n)+c*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{n +1} \left (a +\sqrt {a^{2}-4 b}\right )}{2 n +2}} x \left (\operatorname {KummerM}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.492 (sec). Leaf size: 333
DSolve[y''[x]+a*x^n*y'[x]+(b*x^(2*n)+c*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} \exp \left (-\frac {1}{2} x^{n+1} \left (\frac {\sqrt {a^2-4 b}}{\sqrt {(n+1)^2}}+\frac {a}{n+1}\right )\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )-2 \sqrt {a^2-4 b} c (n+1)}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}},\frac {n}{n+1},\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )+c_2 L_{\frac {2 \sqrt {a^2-4 b} c (n+1)-n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}}}^{-\frac {1}{n+1}}\left (\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )\right ) \]