27.35 problem 45

Internal problem ID [10868]
Internal file name [OUTPUT/10125_Sunday_December_24_2023_05_13_38_PM_31232604/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: 45.
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.

\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y=0} \]

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: 0.407 (sec). Leaf size: 96

dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)+b*x^(n-1)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = x \left (\operatorname {KummerU}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{2} +\operatorname {KummerM}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} \right ) {\mathrm e}^{-\frac {a x \,x^{n}}{n +1}} \]

Solution by Mathematica

Time used: 0.143 (sec). Leaf size: 120

DSolve[y''[x]+a*x^n*y'[x]+b*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \left (\frac {1}{n}+1\right )^{-\frac {1}{n+1}} n^{-\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{n a+a},\frac {n+2}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{n a+a},\frac {n}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right ) \]