49.9.4 problem 1(d)
Internal
problem
ID
[7653]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
2.
Linear
equations
with
constant
coefficients.
Page
83
Problem
number
:
1(d)
Date
solved
:
Wednesday, March 05, 2025 at 04:49:44 AM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\left (5\right )}+2 y&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 136
ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_{1} {\mathrm e}^{-\frac {x \left (i 2^{{7}/{10}} \sqrt {5-\sqrt {5}}-2^{{1}/{5}} \left (\sqrt {5}+1\right )\right )}{4}}+c_{2} {\mathrm e}^{-\frac {x \left (i 2^{{7}/{10}} \left (\sqrt {5}+1\right ) \sqrt {5-\sqrt {5}}+2 \left (\sqrt {5}-1\right ) 2^{{1}/{5}}\right )}{8}}+c_3 \,{\mathrm e}^{-2^{{1}/{5}} x}+c_4 \,{\mathrm e}^{\frac {x \left (i 2^{{7}/{10}} \left (\sqrt {5}+1\right ) \sqrt {5-\sqrt {5}}-2 \left (\sqrt {5}-1\right ) 2^{{1}/{5}}\right )}{8}}+c_5 \,{\mathrm e}^{2^{{1}/{5}} \left (\cos \left (\frac {\pi }{5}\right )+i \sin \left (\frac {\pi }{5}\right )\right ) x}
\]
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 180
ode=D[y[x],{x,5}]+2*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-\frac {\left (\sqrt {5}-1\right ) x}{2\ 2^{4/5}}} \left (c_5 e^{\frac {\left (\sqrt {5}-5\right ) x}{2\ 2^{4/5}}}+c_3 e^{\frac {\sqrt {5} x}{2^{4/5}}} \cos \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_4 \cos \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_2 e^{\frac {\sqrt {5} x}{2^{4/5}}} \sin \left (\frac {\sqrt {5-\sqrt {5}} x}{2\ 2^{3/10}}\right )+c_1 \sin \left (\frac {\sqrt {5+\sqrt {5}} x}{2\ 2^{3/10}}\right )\right )
\]
✓ Sympy. Time used: 1.445 (sec). Leaf size: 563
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*y(x) + Derivative(y(x), (x, 5)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {\sqrt [5]{2} x \left (1 - \sqrt {5}\right )}{4}} \sin {\left (\frac {2^{\frac {7}{10}} x \left (1 + \sqrt {5}\right ) \sqrt {5 - \sqrt {5}}}{8} \right )} + C_{2} e^{\frac {\sqrt [5]{2} x \left (1 - \sqrt {5}\right )}{4}} \cos {\left (\frac {2^{\frac {7}{10}} x \left (1 + \sqrt {5}\right ) \sqrt {5 - \sqrt {5}}}{8} \right )} + C_{3} e^{\frac {\sqrt [5]{2} x \left (1 + \sqrt {5}\right )}{4}} \sin {\left (\frac {2^{\frac {7}{10}} x \sqrt {5 - \sqrt {5}}}{4} \right )} + C_{4} e^{\frac {\sqrt [5]{2} x \left (1 + \sqrt {5}\right )}{4}} \cos {\left (\frac {2^{\frac {7}{10}} x \sqrt {5 - \sqrt {5}}}{4} \right )} + C_{5} e^{\frac {\sqrt [5]{2} x \left (2 + \sqrt {5 - \sqrt {5}} \sqrt {\sqrt {5} + 5}\right )}{8}} \sin {\left (\frac {2^{\frac {7}{10}} x \left (- \sqrt {5} \sqrt {5 - \sqrt {5}} + \sqrt {5 - \sqrt {5}} + \sqrt {\sqrt {5} + 5} + \sqrt {5} \sqrt {\sqrt {5} + 5}\right )}{16} \right )} + C_{6} e^{\frac {\sqrt [5]{2} x \left (2 + \sqrt {5 - \sqrt {5}} \sqrt {\sqrt {5} + 5}\right )}{8}} \cos {\left (\frac {2^{\frac {7}{10}} x \left (- \sqrt {5} \sqrt {5 - \sqrt {5}} + \sqrt {5 - \sqrt {5}} + \sqrt {\sqrt {5} + 5} + \sqrt {5} \sqrt {\sqrt {5} + 5}\right )}{16} \right )} + C_{7} e^{\frac {\sqrt [5]{2} x \left (- \sqrt {5 - \sqrt {5}} \sqrt {\sqrt {5} + 5} + 2\right )}{8}} \sin {\left (\frac {2^{\frac {7}{10}} x \left (- \sqrt {5 - \sqrt {5}} + \sqrt {\sqrt {5} + 5} + \sqrt {5} \sqrt {5 - \sqrt {5}} + \sqrt {5} \sqrt {\sqrt {5} + 5}\right )}{16} \right )} + C_{8} e^{\frac {\sqrt [5]{2} x \left (- \sqrt {5 - \sqrt {5}} \sqrt {\sqrt {5} + 5} + 2\right )}{8}} \cos {\left (\frac {2^{\frac {7}{10}} x \left (- \sqrt {5 - \sqrt {5}} + \sqrt {\sqrt {5} + 5} + \sqrt {5} \sqrt {5 - \sqrt {5}} + \sqrt {5} \sqrt {\sqrt {5} + 5}\right )}{16} \right )} + C_{9} e^{- \sqrt [5]{2} x}
\]