11.2.8 problem 15
Internal
problem
ID
[1479]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
11th
ed.,
Boyce,
DiPrima,
Meade
Section
:
Chapter
4.2,
Higher
order
linear
differential
equations.
Constant
coefficients.
page
180
Problem
number
:
15
Date
solved
:
Tuesday, March 04, 2025 at 12:36:32 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 47
ode:=diff(diff(diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x),x),x)+8*diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (\left (c_4 x +c_2 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (x c_3 +c_1 \right )\right ) {\mathrm e}^{-x}+\left (\left (c_8 x +c_6 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_7 x +c_5 \right )\right ) {\mathrm e}^{x}
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 238
ode=D[y[x],{x,8}]+8*D[y[x],{x,4}]+3*D[y[x],{x,3}]+16*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,2\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,5\right ]\right )+c_6 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,6\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,4\right ]\right )+c_7 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,7\right ]\right )+c_8 \exp \left (x \text {Root}\left [\text {$\#$1}^8+8 \text {$\#$1}^4+3 \text {$\#$1}^3+16\&,8\right ]\right )
\]
✓ Sympy. Time used: 0.213 (sec). Leaf size: 42
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(16*y(x) + 8*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 8)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) \sin {\left (x \right )} + \left (C_{3} + C_{4} x\right ) \cos {\left (x \right )}\right ) e^{- x} + \left (\left (C_{5} + C_{6} x\right ) \sin {\left (x \right )} + \left (C_{7} + C_{8} x\right ) \cos {\left (x \right )}\right ) e^{x}
\]