Question

Find the general solutions of the following equations and compare computer solutions.

${\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is

$y=(C_{1}x+C_3)e^{-x}+(C_{3}x+C_4)e^x+(C_{5}x+C_6)e^{-ix}+(C_{7}x+C_8)e^{ix}$

Step-by-step solution

Given information from question 

Given equation is${(D^4-1)}^{2}y=0$

Differential equation

A differential equation is a formula that connects the derivatives of one or more unknown functions. Functions are used to represent physical quantities, derivatives are used to characterise their rates of change, and differential equations are used to define a relationship between them in applications.

 Step 3: Calculate the general solution

Rewrite the auxiliary equation

$(D+1)(D+1)(D-1)(D-1)(D+i)(D+i)(D-i)(D-i)y=0.$

Which is the eighth order differential equation with four different roots.

To solve this equation, start by taking the simpler equation,

$\begin{array}{c}\frac{dy}{dx}=-y\\ y=b_1{e}^x\end{array}$

Then, to find the second solution form${(D+1)}^{2}y=0$, assume$(D+1)y=u$, therefore it will become. $(D+1)u=0$Thus,

$\begin{array}{c}\frac{du}{dx}=-u\\ u=b_2{e}^{-x}\end{array}$

Calculate first order linear differential equation

Substitute$(D+1)y=u$into the solution, so get a first order linear differential equation

$\begin{array}{c}\frac{dy}{dx}+y=b_2{e}^{-x}\\ I=\int dx=x\\ e^l=e^x\end{array}$

Solve further

$\begin{array}{c}y{e}^l=\int \left(b_2{e}^{-x}\right)e^{x}dx\\ =b_{2}x+C_2\\ y=(C_{1}x+C_3)e^{-x}\end{array}$

Therefore, the solution is

$\begin{array}{c}(D+1)(D+1)y=0\\ y=(C_{1}x+C_3)e^{-x}\end{array}$

The three roots of the equation 

Use the same method for the three roots

$\begin{array}{c}(D-1)(D-1)y=0\\ y=(C_{3}x+C_4)e^x\\ (D+i)(D+i)y=0\\ y=(C_{5}x+C_6)e^{-ix}\end{array}$

Solve further the equation

$\begin{array}{c}(D-i)(D-i)y=0\\ y=(C_{7}x+C_8)e^{ix}\end{array}$

The general solution is a linear combination of all of these independent solutions

$y=(C_{1}x+C_3)e^{-x}+(C_{3}x+C_4)e^x+(C_{5}x+C_6)e^{-ix}+(C_{7}x+C_8)e^{ix}$