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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 [2019-10-08 Tue 11:29] Source Modes and the Characteristic Equation | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 In this system, when is the equilibrium point and what does it mean? 1.1 Front In this system, when is the equilibrium point and what does it mean? 1.2 Back It’s at \(x=0\), and it’s the point where the spring is relaxed, which means that it is exerting no force....

Captured On [2019-10-11 Fri 17:46] Source [[https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/index.htm][Damped Harmonic Oscillators | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare]] 1 What is a damped sinusoid equation? 1.1 Front What is a damped sinusoid equation? Write the general form 1.2 Back \(y(t) = A e^{- \lambda t} \cos(\omega t - \phi)\) 2 What is the basic solutions of DE is we get \(z(t)\) with complex exponential?...

Captured On [2019-10-16 Wed 13:53] Source Exponential Resonse 1 Can we use the superposition principle for second order equation? 1.1 Front Can we use the superposition principle for second order equation? i.e. \(m\ddot{x} + b\dot{x} + kx = F_{ext}\) 1.2 Back Yes, you can. Suppose \(x_p\) is any solution to this equation, and \(x_h\) is the solution of the homogeneous second order equation (\(m\ddot{x} + b\dot{x} + kx = 0\))...

Captured On [2019-10-25 Fri 12:58] Source Gain and Phase Lag | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 When can we say that a system is stable? 1.1 Front When can we say that a system is /stable/? 1.2 Back If the systems’s long-term behaviour does not depend significantly on the initial conditions. 2 Give an example of stable system in mechanics?...

Captured On [2019-10-28 Mon 13:23] Source Undetermined Coefficients | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the undetermined coefficients theorem? 1.1 Front What is the undetermined coefficients theorem? Let \(p(D)y = q(x)\), where \(q(x)\) is a polynomial 1.2 Back \(q(x)\) is a polynomial with the form \(q(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0\), where the largest \(k\) is the degree of polynomial which \(a_k \neq 0\)...

Captured On [2019-10-29 Tue 13:41] Source Linear Operators, Linear Time Invariance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the name of this symbol \(p(D)\)? 1.1 Front What is the name of this symbol $p(D)$? Where \(D\) is a differential operator 1.2 Back \(p(D)\) is a polynomial operator 2 What is a polynomial differential operator with constant coefficients?...

Captured On [2019-10-31 Thu 13:31] Source Pure Resonance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare

Captured On [2019-11-11 Mon 12:39] Source Frequency Response and Practical Resonance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the amplitude response of the system? 1.1 Front What is the amplitude response of the system? 1.2 Back It’s called to the gain of the system in terms of input angular frequency 2 What is the phase response of the system?...

Captured On [2019-11-14 Thu 13:11] Source Applications: LRC Circuits | Unit II: Second Order Constant Coefficient Linear 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?...