71.10.10 problem 10

Internal problem ID [14426]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.3, page 210
Problem number : 10
Date solved : Saturday, February 22, 2025 at 03:47:46 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: 0.005 (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);
 

\[ y = \left (\left (c_4 x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right )\right ) {\mathrm e}^{-x}+\left (\left (c_8 x +c_6 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_7 x +c_5 \right )\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 66

ode=D[y[x],{x,8}]+8*D[y[x],{x,4}]+16*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \left (\left (c_4 x+c_7 e^{2 x}+c_8 e^{2 x} x+c_3\right ) \cos (x)+\left (c_2 x+c_5 e^{2 x}+c_6 e^{2 x} x+c_1\right ) \sin (x)\right ) \]

✓ Sympy. Time used: 0.229 (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)
 

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