Internal problem ID [9904]
Internal file name [OUTPUT/8851_Monday_June_06_2022_05_40_26_AM_95566812/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1582.
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.
\[ \boxed {y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 5 \\ {} & {} & \frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime } \end {array} \]
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 trying reduction of order using simple exponentials --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(y(x),x$5)+a*x^nu*diff(y(x),x)+a*nu*x^(nu-1)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 10.169 (sec). Leaf size: 528
DSolve[y'''''[x]+a*x^\[Nu]*y'[x]+a*\[Nu]*x^(\[Nu]-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \nu ^{-\frac {16}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{-\frac {16}{\nu +4}} a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (c_5 a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \, _1F_4\left (1;\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4},\frac {\nu }{\nu +4}+\frac {7}{\nu +4},\frac {\nu }{\nu +4}+\frac {8}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )+c_4 \nu ^{\frac {4}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {4}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4},\frac {\nu }{\nu +4}+\frac {7}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_3 \nu ^{\frac {8}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {8}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {3}{\nu +4},\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_2 \nu ^{\frac {12}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {12}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {2}{\nu +4},\frac {\nu }{\nu +4}+\frac {3}{\nu +4},\frac {\nu }{\nu +4}+\frac {5}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_1 \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {1}{\nu +4},\frac {\nu }{\nu +4}+\frac {2}{\nu +4},\frac {\nu }{\nu +4}+\frac {3}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right ) \]