5.2.8 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, September 30, 2025 at 04:34:30 AM
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: 0.002 (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);
 

\[ y = \left (\left (c_8 x +c_6 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_7 x +c_5 \right )\right ) {\mathrm e}^{-x}+\left (\left (c_4 x +c_2 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_3 x +c_1 \right )\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.002 (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]
 

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.126 (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)
 

\[ y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) \sin {\left (x \right )} + \left (C_{3} + C_{4} x\right ) \cos {\left (x \right )}\right ) e^{- x} + \left (\left (C_{5} + C_{6} x\right ) \sin {\left (x \right )} + \left (C_{7} + C_{8} x\right ) \cos {\left (x \right )}\right ) e^{x} \]