Let 𝐗=( x y ) . Then 𝐗^'=( 3 -5 4 8 ) 𝐗

Name: Problem 1, Chapter 10: Advanced Engineering Mathematics
Uploaded: 2020-02-04T23:48:22-08:00
Duration: 4 min
Channel: Todd Vawdrey
Description: Let 𝐗=( x y ) . Then 𝐗^'=( 3 -5 4 8 ) 𝐗

Let 𝐗=( x y ) . Then 𝐗^'=( 3 -5 4 ...

Key Concepts

System of Linear Differential Equations

This refers to a set of differential equations in which each equation involves derivatives of several dependent variables and each derivative is expressed as a linear combination of the variables. Such systems are fundamental in modeling situations where multiple quantities interact dynamically and change over time.

Matrix Representation of Differential Systems

Expressing the system in the form X' = AX, where X is the vector of functions and A is the coefficient matrix, provides a compact and powerful framework. This format allows one to leverage linear algebra techniques, such as finding eigenvalues and eigenvectors, to study the system's behavior and solve the differential equations efficiently.

Eigenvalues and Eigenvectors

These are key tools used to analyze the dynamics of a linear differential system. The eigenvalues indicate the growth, decay, or oscillatory behavior of the system, while the eigenvectors determine the directions in which these behaviors occur. Together, they facilitate the process of solving the system and understanding its long-term stability.

Phase Plane Analysis

For systems with two variables, phase plane analysis is used to graphically represent the trajectories of the system. This technique helps in visualizing the behavior of solutions over time, identifying equilibrium points, and inspecting stability and the qualitative nature of the dynamics, such as whether trends are spirals, nodes, or saddles.

Transcript

00:01 Okay, for this derivative, so we're going to take a look at the structure right here, and we're going to get one thing, and we're going to multiply that by another thing.

00:10 So that thing says we're going to use the power rule, and in each of those powers, we're going to actually have to use the product rule, and we're going to have to use the chain rule.

00:23 So product rule tells us we're going to take the derivative of the first thing, which is going to be four times four.

00:30 4x plus 5 to the third times the derivative of the inside of that, which is 4 times just the regular old second, 2x plus 5 to the negative second.

00:45 And then we're going to add the first thing, which is 4x plus 5 to the fourth times the derivative of the second, which is going to be negative 2 times 2x plus 5 to 4x ,000.

01:00 5 to the negative third times the derivative of the inside of that, which is just 2 there.

01:08 Okay, now we're going to go ahead and try and simplify all of these things.

01:14 So the first thing, let's just go ahead and pull common factors together.

01:22 So i have the 4 and the 4, which is 16 times 4x plus 5 to the 3 times 2x ,000.

01:31 Plus 5 to the negative second plus.

01:36 Now i have the negative 4.

01:38 Let's go ahead and put that rather than a plus sign here.

01:42 So negative 4 times 4x plus 5 to the 4th times 2x plus 5 to the negative 3rd.

01:53 Okay, so this piece right here i can see kind of lines up pretty nicely with this.

02:01 Thing right here.

02:02 So i'm going to go ahead and pull out negative three of them.

02:06 What that's going to do is that's going to put it in the denominator as 2x plus 5 being cubed right here...

Please give Ace some feedback