2.8.5 Problem 5

Internal problem ID [18513]
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 : Monday, March 31, 2025 at 05:40:56 PM
CAS classification : [[_high_order, _missing_x]]

Solved as higher order constant coeff ode

Time used: 0.037 (sec)

The characteristic equation is

\[ -a^{4}+\lambda ^{4} = 0 \]

The roots of the above equation are

\begin{align*} \lambda _1 &= a\\ \lambda _2 &= -a\\ \lambda _3 &= i a\\ \lambda _4 &= -i a \end{align*}

Therefore the homogeneous solution is

\[ y_h(x)={\mathrm e}^{a x} c_1 +{\mathrm e}^{-a x} c_2 +{\mathrm e}^{i a x} c_3 +{\mathrm e}^{-i a x} c_4 \]

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

\begin{align*} y_1 &= {\mathrm e}^{a x}\\ y_2 &= {\mathrm e}^{-a x}\\ y_3 &= {\mathrm e}^{i a x}\\ y_4 &= {\mathrm e}^{-i a x} \end{align*}

✓ Maple. Time used: 0.005 (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);
 

\[ y = c_1 \,{\mathrm e}^{-a x}+c_2 \,{\mathrm e}^{a x}+c_3 \sin \left (a x \right )+c_4 \cos \left (a x \right ) \]

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)
 

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