27.44 problem 54

Internal problem ID [10877]
Internal file name [OUTPUT/10134_Sunday_December_24_2023_05_14_03_PM_1011614/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: 54.
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.

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

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) 
   <- linear symmetries successful`
 

✓ Solution by Maple

Time used: 0.484 (sec). Leaf size: 78

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

\[ y \left (x \right ) = -\left (x +c \right ) \left (\left (\int \frac {{\mathrm e}^{-\frac {\left (a \,x^{2} \left (n +2\right ) \left (n +1\right )+\left (a c +b \right ) x \left (3+n \right ) \left (n +1\right )+b c \left (3+n \right ) \left (n +2\right )\right ) x^{n +1}}{\left (3+n \right ) \left (n +1\right ) \left (n +2\right )}}}{\left (x +c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \]

✗ Solution by Mathematica

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

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

Not solved