Consider the following differential equation.ÿ + 3y + 2y = 0state space description ?b) After Solving for x (t)program to plot x(t) 'ist.=).Also plat x₂(t) so x₁ (+).d) Repeat. a), b) ¢ ¢) ific the differential equatioÿ + w₂² y == o" and 2/0 = [b](0)x(0)physical explanation to above resultsGive anwrite a computer- (x²)-[•])

Answered: Consider the ÿ + 3y + 2y = following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Consider the following differential equation.
ÿ + 3y + 2y = 0
state space description ?
b) After Solving for x (t)
program to plot x(t) 'ist.
=).
Also plat x₂(t) so x₁ (+).
d) Repeat. a), b) ¢ ¢) ific the differential equatio
ÿ + w₂² y == o
" and 2/0 = [b]
(0)
x(0)
physical explanation to above results
Give an
write a computer
- (x²)-[•])

Expert Solution

Step 1: (a). The state-space representation:

To represent the given second-order homogeneous differential equation,

$y apostrophe apostrophe space plus space 3 y apostrophe space plus space 2 y space equals space 0$

in state-space form, we need to introduce two first-order differential equations.

The state-space representation is a common way to describe a system of linear differential equations. Here's how you can do it:

Let's define two state variables, x₁ and x_2, which represent y and y', respectively:

$x subscript 1 space equals space y space x subscript 2 space equals space y to the power of apostrophe$

Now, we'll express the derivatives of these state variables in terms of the original equation,

$x subscript 1 apostrophe space equals space y apostrophe space left parenthesis b y space d e f i n i t i o n right parenthesis space x subscript 2 apostrophe space equals space y apostrophe apostrophe space left parenthesis b y space d e f i n i t i o n right parenthesis$

Now, we can rewrite the original differential equation in terms of x₁ and x₂:

$x subscript 1 apostrophe space equals space x subscript 2 space x subscript 2 apostrophe space equals space minus 2 x subscript 1 space minus space 3 x subscript 2$

These two first-order equations, along with the initial conditions, form the state-space representation of the given second-order differential equation.

This system of first-order differential equations can be represented in matrix form as follows:

$fraction numerator d x over denominator d t end fraction equals A x$

Where:

x is the state vector [x₁, x₂]

dx/dt is the derivative of the state vector with respect to time

A is the coefficient matrix:

$A equals open square brackets table row 0 1 row cell negative 2 end cell cell negative 3 end cell end table close square brackets$

This is the state-space description of the given differential equation y'' + 3y' + 2y = 0.