Internal problem ID [10831]
Internal file name [OUTPUT/9813_Sunday_June_19_2022_09_26_04_PM_77415254/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 Equations Containing
Power Functions. page 213
Problem number: 8.
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.
\[ \boxed {y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y=0} \]
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: 0.203 (sec). Leaf size: 136
dsolve(diff(y(x),x$2)-a*(a*x^(2*n)+n*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{2} x^{-\frac {3 n}{2}-1} \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )}{2}+\left (\left (\frac {n}{2}+1\right ) x^{-\frac {3 n}{2}-1}+a \,x^{-\frac {n}{2}}\right ) \left (n +1\right ) c_{2} \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )+c_{1} {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}} \]
✓ Solution by Mathematica
Time used: 0.596 (sec). Leaf size: 81
DSolve[y''[x]-a*(a*x^(2*n)+n*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {a x^{n+1}}{n+1}} \left (c_2-\frac {c_1 2^{-\frac {1}{n+1}} x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {2 a x^{n+1}}{n+1}\right )}{n+1}\right ) \]