60.6.10 problem 1587
Internal
problem
ID
[11586]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
5,
linear
fifth
and
higher
order
Problem
number
:
1587
Date
solved
:
Tuesday, January 28, 2025 at 06:06:57 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime \prime \prime }-a y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 164
dsolve(x^2*diff(y(x),x$4)-a*y(x)=0,y(x), singsol=all)
\[
y = -\frac {\left (\left (-\operatorname {BesselY}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_4 -\operatorname {BesselJ}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_3 \right ) a^{{1}/{4}}-\sqrt {-\sqrt {a}}\, \left (\operatorname {BesselY}\left (1, 2 a^{{1}/{4}} \sqrt {x}\right ) c_{2} +\operatorname {BesselJ}\left (1, 2 a^{{1}/{4}} \sqrt {x}\right ) c_{1} \right )\right ) \sqrt {x}+\left (\operatorname {BesselJ}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_3 +\operatorname {BesselY}\left (0, 2 a^{{1}/{4}} \sqrt {x}\right ) c_{2} +\operatorname {BesselJ}\left (0, 2 a^{{1}/{4}} \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_4 \right ) a^{{1}/{4}} x \sqrt {-\sqrt {a}}}{a^{{1}/{4}} \sqrt {-\sqrt {a}}}
\]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 121
DSolve[x^2*D[y[x],{x,4}]-a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {a x^2}{16}| \begin {array}{c} 0,1,\frac {1}{2},\frac {3}{2} \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {a x^2}{16}| \begin {array}{c} \frac {1}{2},\frac {3}{2},0,1 \\ \end {array} \right )+\frac {1}{64} \sqrt {a} x \left ((4 c_3-3 i c_1) \operatorname {BesselJ}\left (2,2 \sqrt [4]{a} \sqrt {x}\right )+(3 i c_1+4 c_3) \operatorname {BesselI}\left (2,2 \sqrt [4]{a} \sqrt {x}\right )\right )
\]