Internal problem ID [11074]
Internal file name [OUTPUT/10331_Wednesday_January_24_2024_10_07_29_PM_64080441/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 2.1.2-8. Other equations.
Problem number: 251.
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.
\[ \boxed {x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (-a +1\right ) x^{n -1} y=0} \]
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 65
dsolve(x*(x^n+1)*diff(y(x),x$2)+((a-b)*x^n+a-n)*diff(y(x),x)+b*(1-a)*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (x^{-a +n +1} c_{2} \operatorname {hypergeom}\left (\left [\frac {b +n}{n}, \frac {-a +n +1}{n}\right ], \left [\frac {2 n -a +1}{n}\right ], -x^{n}\right )+c_{1} \right ) \left (x^{n}+1\right )^{\frac {b}{n}} \]
✓ Solution by Mathematica
Time used: 0.164 (sec). Leaf size: 69
DSolve[x*(x^n+1)*y''[x]+((a-b)*x^n+a-n)*y'[x]+b*(1-a)*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \left (x^n\right )^{\frac {-a+n+1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {-a-b+n+1}{n},\frac {-a+2 n+1}{n},-x^n\right )+c_1 \left (x^n+1\right )^{b/n} \]