27.7 problem 17

Internal problem ID [10840]
Internal file name [OUTPUT/9822_Sunday_June_19_2022_09_26_17_PM_68406290/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: 17.
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.

\[ \boxed {y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y=0} \]

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 solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> 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: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying 2nd order exact linear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying 2nd order, integrating factor of the form mu(x,y) 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   <- linear_1 successful 
   <- 2nd order, integrating factors of the form mu(x,y) successful`
 

✓ Solution by Maple

Time used: 0.187 (sec). Leaf size: 51

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

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

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[y''[x]+a*y'[x]+b*(-b*x^(2*n)-a*x^n+n*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved