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PROFESSOR: Welcome back. So in this session, we're going to look at unit step and impulse responses. So in this question, we ask you to find the unit impulse response to these two equations, x dot plus 2x equals f of t, and 2 x dot dot plus 27 x dot-- oops, it should be a 7-- plus 7 x dot plus 3x equals f of t. In the second part, you're asked to find the unit step response for the first equation. So here, the key points are really to remember what do we mean by unit impulse and unit step response. Which initial condition correspond to these responses? And what functions of f of t do you choose in each case? So why don't you pause the video and work through this problem? And I'll be right back. Welcome back. So let's look at the equation A. So the unit impulse response is simply-- I'm going to write this down, unit impulse response-- is simply the solution to the following problem, to our differential equation, x dot plus 2x that we're given, w...

PROFESSOR: Hi everyone. Welcome back. So today, I'd like to tackle a problem in undetermined coefficients, specifically find a particular solution to each of the following equations using undetermined coefficients. So for part A, we have x dot plus 3x equals t squared plus t. And for B, we have x dot dot plus x dot equals t to the four. So I'll let you work this problem out. And I'll come back in a minute. Hi Everyone. Welcome back. So we're asked to solve this problem using the method of undetermined coefficients. And specifically, the observation is if we have a differential equation with constant coefficients, and we have a forcing on the right-hand side which is a polynomial, then there's always going to be a particular solution, which is a polynomial, that has the form of some constant times t to the power of r plus constant t^(r-1) plus a constant c_(r-2) t^(r-2) plus dot, dot, dot, plus c_1*t and then possibly plus c_0. And typically, the proble...

PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to talk about positive definite matrices. And specifically, we're going to analyze several properties of positive definite matrices. And specifically, we're going to look at why each one of these following statements is true. So first off, why every positive definite matrix is invertible. Why the only positive definite projection matrix is the identity matrix. If D is a diagonal matrix with positive entries, show that it must also be positive definite. And then lastly, if S is a symmetric matrix where the determinant S is bigger than 0, show why this might not necessarily imply that it's positive definite. So I'll let you think about these for a moment. And I'll come back in a second. Hi, everyone. Welcome back. OK. So let's take a look at part A. So part A is asking why every positive definite matrix is invertible. Well, let's just recall that if A is a matrix and if A is invertible...

PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to talk about positive definite matrices. And specifically, we're going to analyze several properties of positive definite matrices. And specifically, we're going to look at why each one of these following statements is true. So first off, why every positive definite matrix is invertible. Why the only positive definite projection matrix is the identity matrix. If D is a diagonal matrix with positive entries, show that it must also be positive definite. And then lastly, if S is a symmetric matrix where the determinant S is bigger than 0, show why this might not necessarily imply that it's positive definite. So I'll let you think about these for a moment. And I'll come back in a second. Hi, everyone. Welcome back. OK. So let's take a look at part A. So part A is asking why every positive definite matrix is invertible. Well, let's just recall that if A is a matrix and if A is invertible...

MARTINA BALAGOVIC: Hi. Welcome back. Today's problem is about solving homogeneous linear systems, A*x equals 0, but it's also an introduction to the next lecture and next recitation section, which are going to be about solving non-homogeneous linear systems, A*x equals b. The problem is fill the blanks type. And it says the set S of all points with coordinates x, y, and z, such that x minus 5y plus 2z equals 9 is a blank in R^3. It is in a certain relation to the other blank S_0 of all the points with coordinates x, y, and z that satisfy the following linear equation, x minus 5y plus 2z equals 0. After we solve this, we have the second part of the problem, which says all points of x have a specific form, x, y, z equals blank, 0, 0, plus some parameter times blank, 1, 0 plus some other parameter times blank, 0, 1. And we need to fill out all six blanks. Now you should pause the video, fill in the blanks, and then come back and see some pretty pictures that I prepar...

MARTINA BALAGOVIC: Hi. Welcome back. Today's problem is about solving homogeneous linear systems, A*x equals 0, but it's also an introduction to the next lecture and next recitation section, which are going to be about solving non-homogeneous linear systems, A*x equals b. The problem is fill the blanks type. And it says the set S of all points with coordinates x, y, and z, such that x minus 5y plus 2z equals 9 is a blank in R^3. It is in a certain relation to the other blank S_0 of all the points with coordinates x, y, and z that satisfy the following linear equation, x minus 5y plus 2z equals 0. After we solve this, we have the second part of the problem, which says all points of x have a specific form, x, y, z equals blank, 0, 0, plus some parameter times blank, 1, 0 plus some other parameter times blank, 0, 1. And we need to fill out all six blanks. Now you should pause the video, fill in the blanks, and then come back and see some pretty pictures that I prepar...

PROFESSOR: Welcome back. In this session, we're going to tackle initial value problem y dot plus t*y equals to t. And this initial value problem is going to be subject to the initial condition, y of 0 equals to 3. We are going to use the method of integration factor. And what I want you to use is both definite integrals and indefinite integrals. So why don't you take a few minutes to think about the problem? And we will be right back. Welcome back. So I hope that you worked out the first part of the problem. So what are we going to do to solve this ODE? First, we need to review what is the method of integrating factor. So when we use the integrating factor, basically, we're trying to write down our ODE in a different form by introducing a function u. And the goal is to find the function u such that we can rewrite this left-hand side as the derivative of the product y dot u. So in this case, if we're looking at identifying the function u that would give us ...

PROFESSOR: Welcome back. So in this session, we're going to look at sinusoidal inputs for ordinary differential equations of order one. So here in question one, you're asked to use complex techniques to solve x dot plus k*x equals cosine of omega*t. Here k and omega are constants. This is of ODE of first order. And the sinusoidal input is referring to the right-hand side of the ODE where basically the we are forcing a system with a function, and it's sinusoidal of angular frequency omega, which means basically a period of 2pi over omega. So you're asked to use complex techniques to solve this in question a. In question b, we're asked to use what we had in question a to solve this modified function, where again, we have a sinusoidal input on the right-hand side with an additional amplitude F, which is now constant. In the third part, we're asked to use superposition principle to solve this combined equation, where now we're introducing a value f...

PROFESSOR: Welcome back. So today we're going to take a look at a problem with partial fractions, and specifically how to use these partial fractions to compute Laplace inverses. So just as a warm-up, we're asked to recall what the formula is for the Laplace transform of f prime, in terms of the Laplace transform of f. In the second part, we're asked to find the inverse Laplace transform for three different functions. The first one is 1 over s squared minus 4, s squared divided by s squared plus 4, e to the minus 5s divided by s squared minus 4. And notice how none of these functions appear in the look-up table for Laplace transform. So in each case, we have to use partial fractions to convert it or massage these functions into something which we do know the Laplace transform inverse for. And then lastly, for the third problem we're asked to write down the partial fraction decomposition of this function, 1 over s squared times s squared plus 4 times s plus...

PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to tackle a problem in Markov matrices. Specifically, we're going to start with this problem which almost has a physics origin. If we have a particle that jumps between positions A and B with the following probabilities-- I'll just state it-- if it starts at A and jumps to B with probability 0.4 or starts at A and stays at A with probability 0.6, or if it starts at B then it goes to A with probability 0.2 or stays at B with probability 0.8, we'd like to know the evolution of the probability of this particle over a long period of time. So specifically the problem we're interested today is: if we have a particle and we know that it starts at position A, what is the probability that it is at position A and the probability that it's at position B after one step, after n steps, and then finally after an infinite number of steps? So I'll let you think about this problem for a moment and I'...

PROFESSOR: Hi, everyone. Welcome back. So today, I'd like to tackle a problem in Markov matrices. Specifically, we're going to start with this problem which almost has a physics origin. If we have a particle that jumps between positions A and B with the following probabilities-- I'll just state it-- if it starts at A and jumps to B with probability 0.4 or starts at A and stays at A with probability 0.6, or if it starts at B then it goes to A with probability 0.2 or stays at B with probability 0.8, we'd like to know the evolution of the probability of this particle over a long period of time. So specifically the problem we're interested today is: if we have a particle and we know that it starts at position A, what is the probability that it is at position A and the probability that it's at position B after one step, after n steps, and then finally after an infinite number of steps? So I'll let you think about this problem for a moment and I'...

PROFESSOR: Hi everyone. Welcome back. So today I'd like to look at a problem on manipulating Fourier series. Specifically, we're asked to find the Fourier series of cosine 2t minus pi/4. And then the second problem is given a square wave function, which takes on the value of minus 1 and 1, and it's 2pi periodic. We're also told that the square wave function has the Fourier series 4/pi, 1/n sine n*t. And the question is to find the Fourier series of three related functions. So the first function takes on the value of 0 and 4 on the intervals minus pi to 0 and 0 to pi. Note that this function is also 2pi periodic. The second problem it is to find the Fourier series of a function which is minus 1 from minus 1 to 0, and 1 from 0 to 1. So this is the square wave function, but now has period two. And then lastly we're asked to find the Fourier series of a function f of t, which is absolute t. And it's defined as absolute t on the interval from minus pi t...

PROFESSOR: Welcome back. So in this session, we're going to use the matrix method to solve this linear system of differential equations. These are x dot equals 6x plus 5y, and y dot equals x plus 2y. So why don't you take a few minutes to write down the system in matrix form and go through the matrix method to solve it. And I'll be right back. Welcome back. So let's write down this system in matrix form. You would have a vector with entries x and y prime equals a matrix with entries 6, 5; 1, 2 multiplying the column vector [x, y]. So now, we did big part of the work. The matrix method tells us that we need to find the eigenvalues of this matrix to be able to basically diagonalize it and seek eigenvectors so that then we can just read off the solutions and write the solution of the system as a linear combination of the eigenvectors that we found. So let's look for the eigenvalues first. The eigenvalues would be computed by seeking the determinant of thi...

PROFESSOR: Hi everyone. Welcome back. So today, I'd like to take a look at solving a linear ODE but with a periodic input. And specifically, we're asked to find one solution, a particular solution, which is also the periodic solution of the differential equation x dot dot plus 2x dot plus 4x equals the square wave function. So the square wave function is a periodic function with period 2pi. It's defined as minus 1 and 1 on the intervals minus pi to 0 and 0 to pi. And we know that the square wave function has the following Fourier series. So I'll let you think about this problem for a moment. And I'll come back in a second. Hi everyone. Welcome back. So the reason we've been studying Fourier series is to essentially solve differential equations with complicated forcing inputs on the right-hand side which are periodic. And the reason we've been studying Fourier series is because we know that differential equations with sines and cosines as forcin...