60.6.10 problem 1587
Internal
problem
ID
[11547]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
5,
linear
fifth
and
higher
order
Problem
number
:
1587
Date
solved
:
Sunday, March 30, 2025 at 08:24:39 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime \prime \prime }-a y&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 161
ode:=x^2*diff(diff(diff(diff(y(x),x),x),x),x)-a*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\left (\operatorname {BesselJ}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_3 +\operatorname {BesselY}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_4 \right ) a^{{1}/{4}}+\left (\operatorname {BesselY}\left (1, 2 a^{{1}/{4}} \sqrt {x}\right ) c_2 +\operatorname {BesselJ}\left (1, 2 a^{{1}/{4}} \sqrt {x}\right ) c_1 \right ) \sqrt {-\sqrt {a}}\right ) \sqrt {x}-\sqrt {-\sqrt {a}}\, \left (\operatorname {BesselJ}\left (0, 2 a^{{1}/{4}} \sqrt {x}\right ) c_1 +\operatorname {BesselY}\left (0, 2 a^{{1}/{4}} \sqrt {x}\right ) c_2 +\operatorname {BesselJ}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_3 +\operatorname {BesselY}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_4 \right ) x \,a^{{1}/{4}}}{a^{{1}/{4}} \sqrt {-\sqrt {a}}}
\]
✓ Mathematica. Time used: 0.034 (sec). Leaf size: 121
ode=x^2*D[y[x],{x,4}]-a*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {a x^2}{16}| \begin {array}{c} 0,1,\frac {1}{2},\frac {3}{2} \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {a x^2}{16}| \begin {array}{c} \frac {1}{2},\frac {3}{2},0,1 \\ \end {array} \right )+\frac {1}{64} \sqrt {a} x \left ((4 c_3-3 i c_1) \operatorname {BesselJ}\left (2,2 \sqrt [4]{a} \sqrt {x}\right )+(3 i c_1+4 c_3) \operatorname {BesselI}\left (2,2 \sqrt [4]{a} \sqrt {x}\right )\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*y(x) + x**2*Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -a*y(x) + x**2*Derivative(y(x), (x, 4))