85.45.3 problem 2 (c)
Internal
problem
ID
[22786]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
4.
Linear
differential
equations.
A
Exercises
at
page
180
Problem
number
:
2
(c)
Date
solved
:
Thursday, October 02, 2025 at 09:14:38 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\left (6\right )}-4 y^{\prime \prime }+4 y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 29
ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)-4*diff(diff(y(x),x),x)+4*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 +\left (\moverset {5}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-4 \textit {\_Z} +4, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}}\right )
\]
✓ Mathematica. Time used: 0.049 (sec). Leaf size: 193
ode=D[y[x],{x,6}]-4*D[y[x],{x,2}]+4*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,3\right ]\right )}{\text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,3\right ]}+\frac {c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,2\right ]\right )}{\text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,2\right ]}+\frac {c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,5\right ]\right )}{\text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,5\right ]}+\frac {c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,4\right ]\right )}{\text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,4\right ]}+\frac {c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^5-4 \text {$\#$1}+4\&,1\right ]}+c_6 \end{align*}
✓ Sympy. Time used: 0.527 (sec). Leaf size: 105
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*Derivative(y(x), x) - 4*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 6)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + C_{6} e^{x \operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 0\right )}} + \left (C_{2} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 1\right )}\right )} \right )} + C_{3} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 1\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 1\right )}\right )}} + \left (C_{4} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 3\right )}\right )} \right )} + C_{5} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 3\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - 4 x + 4, 3\right )}\right )}}
\]