Question

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

${\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{-}\mathbf{16}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is$y=(C_{2}x+C_8)e^{-x}+C_4{e}^{2x}+C_5{e}^{-2x}+C_6{e}^{2ix}+C_7{e}^{-2ix}$

Step-by-step solution

 Step 1: Given information from question

Given equation is.${(D+1)}^2(D^4-16)y=0$

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.

Calculate the general solution of (D+1)2(D4−16)y=0 

Rewriting the auxiliary equation

$(D+1)(D+1)(D-2)(D+2)(D-i2)(D-i2)y=0.$

This is the sixth order differential equation which have five different roots, two of them are complex. To solve this equation, start by the simpler equation

$\begin{array}{c}(D+1)y=0\\ \frac{dy}{dx}=-y\\ y=C_1{e}^x\end{array}$

Calculate the second solution

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

$\begin{array}{c}\frac{du}{dx}=-u\\ u=C_2{e}^{-x}\end{array}$

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

$\begin{array}{c}\frac{dy}{dx}+y=C_2{e}^{-x}\\ I=\int dx\\ =x\\ e^I=e^x\end{array}$

Solve further the equation

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

The general solution of the equation 

Take the equations, and, then

$\begin{array}{c}\frac{dy}{dx}=2y\\ y=C_4{e}^{2x}\\ \frac{dy}{dx}=-2y\\ y=C_5{e}^{-2x}\end{array}$

And finally, for$(D-i2)y=0$and,$(D+i2)y=0$then

$\begin{array}{c}\frac{dy}{dx}=i2y\\ y=C_6{e}^{2ix}\\ \frac{dy}{dx}=-i2y\\ y=C_7{e}^{-2ix}\end{array}$

Therefore, the general solution is$y=(C_{2}x+C_8)e^{-x}+C_4{e}^{2x}+C_5{e}^{-2x}+C_6{e}^{2ix}+C_7{e}^{-2ix}$