85.46.2 problem 2 (b)
Internal
problem
ID
[22790]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
4.
Linear
differential
equations.
C
Exercises
at
page
181
Problem
number
:
2
(b)
Date
solved
:
Thursday, October 02, 2025 at 09:14:39 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\left (5\right )}-y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 101
ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (\sqrt {5}+1\right ) x}{4}} \left (-c_2 \sin \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, x}{4}\right )+c_4 \cos \left (\frac {\sqrt {2}\, \sqrt {5-\sqrt {5}}\, x}{4}\right )\right )+c_1 \,{\mathrm e}^{x}+{\mathrm e}^{\frac {\left (\sqrt {5}-1\right ) x}{4}} \left (-\sin \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, x}{4}\right ) c_3 +\cos \left (\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, x}{4}\right ) c_5 \right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 154
ode=D[y[x],{x,5}]-y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\frac {1}{4} \left (1+\sqrt {5}\right ) x} \left (c_1 e^{\frac {1}{4} \left (5+\sqrt {5}\right ) x}+c_3 \cos \left (\sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} x\right )+c_2 e^{\frac {\sqrt {5} x}{2}} \cos \left (\sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} x\right )+c_4 \sin \left (\sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} x\right )+c_5 e^{\frac {\sqrt {5} x}{2}} \sin \left (\sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} x\right )\right ) \end{align*}
✓ Sympy. Time used: 0.234 (sec). Leaf size: 143
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x) + Derivative(y(x), (x, 5)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (-1 + \sqrt {5}\right )}{4}} \sin {\left (\frac {\sqrt {2} x \sqrt {\sqrt {5} + 5}}{4} \right )} + C_{2} e^{\frac {x \left (-1 + \sqrt {5}\right )}{4}} \cos {\left (\frac {\sqrt {2} x \sqrt {\sqrt {5} + 5}}{4} \right )} + C_{3} e^{- \frac {x \left (1 + \sqrt {5}\right )}{4}} \sin {\left (\frac {\sqrt {2} x \sqrt {5 - \sqrt {5}}}{4} \right )} + C_{4} e^{- \frac {x \left (1 + \sqrt {5}\right )}{4}} \cos {\left (\frac {\sqrt {2} x \sqrt {5 - \sqrt {5}}}{4} \right )} + C_{5} e^{x}
\]