60.5.22 problem 1557
Internal
problem
ID
[11556]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
4,
linear
fourth
order
Problem
number
:
1557
Date
solved
:
Tuesday, January 28, 2025 at 06:06:50 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 159
dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+8*x*diff(diff(diff(y(x),x),x),x)+12*diff(diff(y(x),x),x)-lambda^2*y(x)=0,y(x), singsol=all)
\[
y = -\frac {-\operatorname {BesselJ}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_3 \sqrt {\lambda }-\operatorname {BesselY}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_4 \sqrt {\lambda }+\sqrt {-\lambda }\, \left (\sqrt {\lambda }\, \sqrt {x}\, \operatorname {BesselJ}\left (0, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_3 +\sqrt {\lambda }\, \sqrt {x}\, \operatorname {BesselY}\left (0, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_4 -c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )-c_{2} \operatorname {BesselY}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )+\sqrt {\lambda }\, \sqrt {x}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {\lambda }\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {\lambda }\, \sqrt {x}\right ) c_{2} \right )\right )}{x^{{3}/{2}} \sqrt {\lambda }\, \sqrt {-\lambda }}
\]
✓ Solution by Mathematica
Time used: 0.040 (sec). Leaf size: 146
DSolve[-(\[Lambda]^2*y[x]) + 12*D[y[x],{x,2}] + 8*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| \begin {array}{c} -1,0,-\frac {1}{2},\frac {1}{2} \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| \begin {array}{c} -\frac {1}{2},\frac {1}{2},-1,0 \\ \end {array} \right )-\frac {3 i c_1 \left (\operatorname {BesselI}\left (2,2 \sqrt {x} \sqrt {\lambda }\right )-\operatorname {BesselJ}\left (2,2 \sqrt {x} \sqrt {\lambda }\right )\right )}{4 \lambda x}-\frac {c_3 \left (\operatorname {BesselJ}\left (2,2 \sqrt {x} \sqrt {\lambda }\right )+\operatorname {BesselI}\left (2,2 \sqrt {x} \sqrt {\lambda }\right )\right )}{\lambda x}
\]