Captured On [2019-09-17 Tue 16:00] Source Basic DE’s and Separable Equations | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 Can be \(e^{at}=0\)? 1.1 Front Can be \(e^{at}=0\)? 1.2 Back Never 2 Write a sketch of the graph 2.1 Front Write a sketch of the graph \(y=e^t\) 2.2 Back 3 Write a sketch of the graph 3.1 Front Write a sketch of the graph...
Captured On [2019-09-17 Tue 16:00] Source Geometric Methods | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a direction field 1.1 Front What is a direction field for the equation \(y’ = f(x,y)\) 1.2 Back For each point \((x,y)\) of the plane is drawn a little segment whose slope is \(f(x,y)\) For example: \({\displaystyle \dv{y}{x} = 2x}\) 2 How can we draw a direction field by hand?...
Captured On [2019-09-11 Wed 20:32] Source First Order Linear ODE’s | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a first order linear ODE? 1.1 Front What is a first order linear ODE? Write the standard form 1.2 Back In the unknown function \(x = x(t)\) \({\displaystyle A(t) \dv{x}{t} + B(t) x(t) = C(t)}\) As \(A(t) \neq 0\), we can simplify the equation by dividing by \(A(t)\)...
Captured On [2019-09-17 Tue 15:58] Source Complex Arithmetic and Exponentials | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a complex number? 1.1 Front What is a complex number? 1.2 Back A complex number is an expression of the form \(a + ib\) 2 When can we say that 2 complex number are equals? 2.1 Front When can we say that 2 complex number are equals?...
Captured On [2019-09-26 Thu 15:47] Source Sinusoidal Functions | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a sinusoidal function equation? 1.1 Front What is a sinusoidal function equation? 1.2 Back \(f(t) = A \cos(\omega t - \phi)\) 2 What is the sinusoidal oscillation equation? 2.1 Front What is the sinusoidal oscillation equation? 2.2 Back \(f(t) = A \cos(\omega t - \phi)\)...
Captured On [2019-09-27 Fri 19:04] Source First Order Constant Coefficient Linear ODE’s | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a constant coefficient First ODE? 1.1 Front What is a constant coefficient First ODE? 1.2 Back \(\dot{y} + ky = q(t)\), where \(k\) is a constant 2 What is the solution of a constant coefficient First ODE? 2.1 Front What is the solution of a constant coefficient First ODE?...
Captured On [2019-10-01 Tue 12:22] Source Exponential Input; Gain and Phase Lag | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 Explain the method of optimism to solve this first order ODE 1.1 Front Explain the method of optimism to solve this first order ODE \(\dot{x} + 2x = 4e^{3t}\) 1.2 Back The inspiration is based on the fact that \(\dv{t} e^{rt} = re^{rt}\)....
Captured On [2019-10-02 Wed 13:06] Source First Order Autonomous Differential Equations | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an autonomous first order differential equation? 1.1 Front What is an autonomous first order differential equation? 1.2 Back There are (in general) nonlinear equations of the form \(\dot{x} = f(x)\) The word autonomous means self governing and indicates that the rate of change of \(x\) is governed by \(x\) itself and it not dependent of time....
Captured On [2019-10-07 Mon 13:29] Source Linear vs. Nonlinear | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare
Captured On [2019-11-15 Fri 13:49] Source Fourier Series: Basics | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 Which is the minimum period of a constant function? 1.1 Front Which is the minimum period of a constant function? 1.2 Back There is no minimal period, we don’t allow \(P = 0\) to be periodic. Also, for any \(P\) value is a period, but no minimal...
Captured On [2019-11-23 Sat 18:11] Source Operations on Fourier Series | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an even function? 1.1 Front What is an even function? Let \(f(t)\) be a function, and write examples 1.2 Back \(f(t)\) is even if \(f(-t) = f(t)\) for all \(t\) \(t^2, t^4, t^6, \dots\), any even power of \(t\) \(\cos(at)\) power series for \(\cos(at)\) has only even powers of \(t\) A constant function 2 What is the integral of even function on a ‘balanced’ interval?...
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...