Internal problem ID [9910]
Internal file name [OUTPUT/8857_Monday_June_06_2022_05_41_09_AM_34874811/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1588.
ODE order: 5.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{10} y^{\left (5\right )}-a y=0} \] Unable to solve this ODE.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 5; missing the dependent variable trying a solution in terms of MeijerG functions <- MeijerG function solution successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 90
dsolve(x^10*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {9}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {8}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {7}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{4} x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {6}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{5} x^{4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}\right ], -\frac {a}{3125 x^{5}}\right ) \]
✓ Solution by Mathematica
Time used: 11.265 (sec). Leaf size: 103
DSolve[x^10*y'''''[x]-a*y[x]== 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^4 \left (c_1 e^{-\frac {\sqrt [5]{a}}{x}}+c_2 e^{\frac {\sqrt [5]{-1} \sqrt [5]{a}}{x}}+c_3 e^{-\frac {(-1)^{2/5} \sqrt [5]{a}}{x}}+c_4 e^{\frac {(-1)^{3/5} \sqrt [5]{a}}{x}}+c_5 e^{-\frac {(-1)^{4/5} \sqrt [5]{a}}{x}}\right ) \]