2.8.5 problem 5

Internal problem ID [18260]
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, December 23, 2024 at 09:24:23 PM
CAS classification : [[_high_order, _missing_x]]

Solve

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

Solved as higher order constant coeff ode

Time used: 0.139 (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 step by step solution

Maple trace

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

Maple dsolve solution

Solving time : 0.007 (sec)
Leaf size : 30

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

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

Mathematica DSolve solution

Solving time : 0.012 (sec)
Leaf size : 53

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

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