Question

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

${\mathit{D}}^{\mathbf{2}}{\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}{\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{3}}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is$y=(c_{1}x+c_2)+(c_{3}x+c_4)e^x+(c_5{x}^2+c_{6}x+c_7)e^{-2x}.$

Step-by-step solution

 Step 1: Given information from question

Given equation is$D^2{(D-1)}^2{(D+2)}^{3}y=0$

Differential equation

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

Rewriting the auxiliary equation

$DD(D-1)(D-1)(D+2)(D+2)(D+2)y=0$

Which is mean that this is seventh order differential equation. It is found that the general solution for any differential that has auxiliary equation with unequal real roots which is,

$y=c_1{e}^{a_{1}x}+c_1{e}^{a_{2}x}+\dots \mathrm{..}+c_n{e}^{a_{n}x}$

Calculate the second solution of differential equation 

Usethe second root, which is $D=-1$, then find the second solution of this differential equation

$\begin{array}{c}(D-a_1)(D-a_2)\dots \dots \mathrm{..}(D-a_n)y=0\\ y=c_2{e}^{-x}\end{array}$

Here. $a_1=a_2=\dots \dots a_n$the solution for such equation is

$y=(c_1{x}^{n-1}+c_2{x}^{n-2}+\dots \mathrm{..}+c_n{x}^{n-n})e^{-(n-1)x}$

Therefore, the solution of $D^{2}y=0$is

$y=c_{1}x+c_2$

The solution of${(D-1)}^{2}y=0$is

$y=(c_{3}x+c_4)e^x$

The solution of ${(D+2)}^{2}y=0$is

$y=(c_5{x}^2+c_6+c_7)e^{-2x}$

And the general solution for our auxiliary equation is a linear combination of the solutions of the simpler equation which is

$y=(c_{1}x+c_2)+(c_{3}x+c_4)e^x+(c_5{x}^2+c_{6}x+c_7)e^{-2x}.$