2.8.5 Problem 5

Internal problem ID [18505]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter VII. Linear equations of order higher than the first. section 56. Problems at page 163
Problem number : 5
Date solved : Thursday, March 13, 2025 at 12:10:54 PM
CAS classification : [[_high_order, _missing_x]]

Solve

$$\begin{array}{rl}{y}^{\prime \prime \prime \prime }-a^{4}y& =0\end{array}$$

Solved as higher order constant coeff ode

Time used: 0.035 (sec)

The characteristic equation is

$$-a^4+{\lambda }^4=0$$

The roots of the above equation are

$$\begin{array}{rl}{\lambda }_1& =a\\ {\lambda }_2& =-a\\ {\lambda }_3& =ia\\ {\lambda }_4& =-ia\end{array}$$

Therefore the homogeneous solution is

$$y_h(x)={\mathrm{e}}^{iax}{c}_1+{\mathrm{e}}^{-iax}{c}_2+{\mathrm{e}}^{-ax}{c}_3+{\mathrm{e}}^{ax}{c}_4$$

The fundamental set of solutions for the homogeneous solution are the following

$$\begin{array}{rl}{y}_1& ={\mathrm{e}}^{iax}\\ y_2& ={\mathrm{e}}^{-iax}\\ y_3& ={\mathrm{e}}^{-ax}\\ y_4& ={\mathrm{e}}^{ax}\end{array}$$

✓ Maple. Time used: 0.008 (sec). Leaf size: 30

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

$$y\left(x\right)=c_1{\mathrm{e}}^{-ax}+c_2{\mathrm{e}}^{ax}+c_3\sin \left(ax\right)+c_4\cos \left(ax\right)$$

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 53

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

$$y(x)\to c_2{e}^{-\sqrt{a}x}+c_4{e}^{\sqrt{a}x}+c_1\cos \left(\sqrt{a}x\right)+c_3\sin \left(\sqrt{a}x\right)$$

✓ Sympy. Time used: 0.131 (sec). Leaf size: 32

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

$$y\left(x\right)=C_1{e}^{-ax}+C_2{e}^{ax}+C_3{e}^{-iax}+C_4{e}^{iax}$$