50.13.9 problem 9

Internal problem ID [8024]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Section 2.7. HIGHER ORDER LINEAR EQUATIONS, COUPLED HARMONIC OSCILLATORS Page 98
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 05:23:27 AM
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: 0.001 (sec). Leaf size: 26

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);
 

\[ y = \left (c_4 x +c_3 \right ) {\mathrm e}^{-a x}+{\mathrm e}^{a x} \left (c_{2} x +c_{1} \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

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]
 

\[ y(x)\to e^{-a x} \left (c_3 e^{2 a x}+x \left (c_4 e^{2 a x}+c_2\right )+c_1\right ) \]

✓ Sympy. Time used: 0.120 (sec). Leaf size: 22

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)
 

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