78.15.9 problem 9

Internal problem ID [18294]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 22. Higher Order Linear Equations. Coupled Harmonic Oscillators. Problems at page 160
Problem number : 9
Date solved : Monday, March 31, 2025 at 05:25:01 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*a^2*diff(diff(y(x),x),x)+a^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_4 x +c_2 \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (c_3 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode=D[y[x],{x,4}]+2*a^2*D[y[x],{x,2}]+a^4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to (c_2 x+c_1) \cos (a x)+(c_4 x+c_3) \sin (a x) \]

✓ Sympy. Time used: 0.127 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**4*y(x) + 2*a**2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- i a x} + \left (C_{3} + C_{4} x\right ) e^{i a x} \]