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Captured On [2019-11-29 Fri 13:34] Source [[https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/odes-with-periodic-input-resonance/index.htm][ODE’s with Periodic Input, Resonance | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare]] 1 How can we solve this ODE? 1.1 Front How can we solve this ODE? \(\ddot{x} + 9.1 x = f(t)\), where \(f(t)\) is a odd square wave of period \(2\pi\) with \(f(t) = 1\) for \(0 \lt t \lt \pi\) 1.2 Back Use the Fourier Series of \(f(t)\) \({\displaystyle f(t) = \frac{4}{\pi} \sum_{n \text{ odd}}^{\infty} \frac{\sin(nt)}{n}}\) So the DE: \({\displaystyle \ddot{x} + 9....

Captured On [2019-12-06 Fri 14:05] Source Step and Delta Functions: Integrals and Generalized Derivatives | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare This session looks closely at discontinuous functions and introduces the notion of an impulse or delta function. The goal is to use these functions as the input to differential equations. Step functions and delta functions are not differentiable in the usual sense, but they do have what we will call generalized derivatives, which are suitable for use in DE’s....

Captured On [2019-12-17 Tue 13:01] Source Unit Step and Unit Impulse Response | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare In this session we study differential equations with step or delta functions as input. For physical systems, this means that we are looking at discontinuous or impulsive inputs to the system. 1 When happens the pre-initial conditions in a DE? 1.1 Front When happens the pre-initial conditions in a DE?...

Captured On [2019-12-27 Fri 13:46] Source Convolution | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the definition of the convolution of 2 functions? 1.1 Front What is the definition of the convolution of 2 functions? One-sided convolution 1.2 Back It’s a function defined by this integral \({\displaystyle (f * g)(t) = \int_{0^-}^{t^+} f(\tau) g(t - \tau) \dd{\tau}}\) for \(t \gt 0\)...

Captured On [2019-12-30 Mon 17:43] Source Laplace Transform: Basics | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the definition of Laplace Transform? 1.1 Front What is the definition of Laplace Transform? 1.2 Back The Laplace transform of a function \(f(t)\) of a real variable \(t\) is another function depending on a new variable \(s\), which is in general complex....

Captured On [2020-01-02 Thu 18:40] Source Partial Fractions and Inverse Laplace Transform | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a rational function? 1.1 Front What is a rational function? 1.2 Back Is a function that is the ratio of two polynomials \({\displaystyle \frac{s + }{s^2 + 7s + 9}}\) 2 When can we say that a rational function is proper?...

Captured On [2020-01-06 Mon 19:03] Source Laplace Transform: Solving Initial Value Problems | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 Compute with the definition the Laplace transform of \(f’\) 1.1 Front Compute with the definition the Laplace transform of $f’$ \({\displaystyle \mathcal{L}(f’)(s)}\) Don’t do by heart 1.2 Back \({\displaystyle \mathcal{L}(f’) = \int_{0^-}^{\infty} f’(t) e^{-st} \dd{t}}\) Integrating by parts \(u = e^{-st}\), \(\dd{u} = -s e^{-st} \dd{t}\) \(\dd{v} = f’(t) \dd{t}\), \(v = f(t)\) \({\displaystyle f(t) e^{-st} \bigg|_{0^-}^{\infty} + s \int_{0^-}^{\infty} f(t) e^{-st} \dd{t} = -f(0^-) + s F(s)}\)...

Captured On [2020-01-10 Fri 13:16] Source Transfer (System) and Weight Functions, Green’s Formula | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the transfer function for any LTI? 1.1 Front What is the transfer function for any LTI? In terms of unit impulse response 1.2 Back \({\displaystyle W(s) = \mathcal{L}(w(t))}\) where \(w(t)\) is the unit impulse response 2 What is the system function?...

Captured On [2020-01-11 Sat 17:01] Source Poles, Amplitude Response, Connection to ERF | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 When can we say that a rational function is in reduced form? 1.1 Front When can we say that a rational function is in reduced form? 1.2 Back If the numerator \(q(s)\) and the denominator \(p(s)\) have no roots in common, then \(q(s)/p(s)\) is in reduced form...

Captured On [2020-01-21 Tue 13:07] Source Linear Systems | Unit IV First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 Can we solve a linear system of ODE’s with constant coefficients by eliminating variables? 1.1 Front Can we solve a linear system of ODE’s with constant coefficients by eliminating variables? 1.2 Back No, it’s a naive way to solve it. You need to use techniques of constant coefficient ODE methods....

Captured On [2020-02-01 Sat 12:35] Source Matrix Methods: Eigenvalues and Normal Modes | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the trace of a square matrix? 1.1 Front What is the trace of a square matrix? For example, \({\displaystyle \operatorname{tr}\begin{pmatrix}a & b \\ c & d\end{pmatrix}}\) 1.2 Back It’s the sum of the elements on the main diagonal; it’s denoted \(\operatorname{tr}(A)\):...

Captured On [2020-02-13 Thu 17:57] Source Qualitative Behavior: Phase Portraits | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the phase plane for this system? 1.1 Front What is the phase plane for this system? \({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \({\displaystyle A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}}\) 1.2 Back It’s the \(xy\text{-plane}\) itself, where you can draw the trajectory of a solution with an arrow to indicate the direction of increasing time....

Captured On [2020-02-20 Thu 21:51] Source Matrix Exponentials | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 For this linear system, how many solutions are there? 1.1 Front For this linear system, how many solutions are there? \({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \(A\) is a \(n \cross n\) matrix 1.2 Back There are \(n\) linearly independent solutions for the system 2 How is the linear system when the coefficients are functions of the independent variable t?...

Captured On [2020-03-06 Fri 14:04] Source Nonlinear Systems | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an autonomous system? 1.1 Front What is an autonomous system? General First Order Autonomous Systems 1.2 Back The word autonomous means self regulating. These systems are self regulating in the sense that their rate of change (e.g. derivatives) depends only on the state of the system (values of \(x\) and \(y\)) and not on the time \(t\)...

Captured On [2020-03-09 Mon 12:02] Source Linearization Near Critical Points | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 How can we get information about the trajectories of autonomous linear system without… 1.1 Front How can we get information about the trajectories of autonomous linear system without… determining them analytically or using a computer to plot them 1.2 Back Analyzing what happens near critical points....

Captured On [2020-03-16 Mon 19:13] Source Limitations of the Linear: Limit Cycles and Chaos | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the form of the solutions that trace out a closed curve? 1.1 Front What is the form of the solutions that trace out a closed curve? In a non-linear system 1.2 Back The solution \(\vb{x}(t)\) will be geometrically realized by a point which goes round and round the curve \(C\) with a certain period \(T\)...