5.2.4 problem 11

Internal problem ID [1475]
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 : 11
Date solved : Tuesday, September 30, 2025 at 04:34:29 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (6\right )}+y&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 56

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (-\sin \left (\frac {x}{2}\right ) c_4 +c_6 \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\sin \left (\frac {x}{2}\right ) c_3 +c_5 \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+c_1 \sin \left (x \right )+c_2 \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 92

ode=D[y[x],{x,6}]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {\sqrt {3} x}{2}} \left (c_1 e^{\sqrt {3} x}+c_3\right ) \cos \left (\frac {x}{2}\right )+c_2 \cos (x)+c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x) \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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