Question

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

$\mathbf{(}{\mathbf{D}}^{\mathbf{3}}\mathbf{+}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{6}\mathbf{D}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is.$y=c_1+c_2{e}^{-3x}+c_3{e}^{2x}$

Step-by-step solution

Given information from question 

Given equation is.$(D^3+D^2-6D)y=0$$(D^3+D^2-6D)y=0$$(D^3+D^2-6D)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(D3+D2−6D)y=0

The auxiliary equation is,

$\begin{array}{l}(D^3+D^2-6D)y=0\\ D(D+3)(D-2)y=0\end{array}$

It is found that the general solution for any differential that has an auxiliary equation with unequal roots is

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

Where in our cases.$a_1=0,a_2=-3,a_3=2$

Then,the differential equation's generic solution is,$y=c_1+c_2{e}^{-3x}+c_3{e}^{2x}$