60.6.5 problem 1582
Internal
problem
ID
[11581]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
5,
linear
fifth
and
higher
order
Problem
number
:
1582
Date
solved
:
Monday, January 27, 2025 at 11:23:42 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y&=0 \end{align*}
✗ 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: 10.105 (sec). Leaf size: 528
DSolve[D[y[x],{x,5}]+a*x^\[Nu]*D[y[x],x]+a*\[Nu]*x^(\[Nu]-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \nu ^{-\frac {16}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{-\frac {16}{\nu +4}} a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \left (c_5 a^{\frac {1}{\nu +4}} \left (x^{\nu }\right )^{\frac {1}{\nu }} \, _1F_4\left (1;\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4},\frac {\nu }{\nu +4}+\frac {7}{\nu +4},\frac {\nu }{\nu +4}+\frac {8}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )+c_4 \nu ^{\frac {4}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {4}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4},\frac {\nu }{\nu +4}+\frac {7}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_3 \nu ^{\frac {8}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {8}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {3}{\nu +4},\frac {\nu }{\nu +4}+\frac {5}{\nu +4},\frac {\nu }{\nu +4}+\frac {6}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_2 \nu ^{\frac {12}{\nu +4}} \left (\frac {\nu +4}{\nu }\right )^{\frac {12}{\nu +4}} \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {2}{\nu +4},\frac {\nu }{\nu +4}+\frac {3}{\nu +4},\frac {\nu }{\nu +4}+\frac {5}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )\right )+c_1 \, _0F_3\left (;\frac {\nu }{\nu +4}+\frac {1}{\nu +4},\frac {\nu }{\nu +4}+\frac {2}{\nu +4},\frac {\nu }{\nu +4}+\frac {3}{\nu +4};-\frac {a \left (x^{\nu }\right )^{\frac {\nu +4}{\nu }}}{(\nu +4)^4}\right )
\]