96
96

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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In the preceding note we discussed the structural stability of a linear system. How does it apply to non-linear systems?

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Nonlinear...

Source: http://www.flooved.com/reader/1437

124
124

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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De�nition: The convolution of two functions f and g is a third function which we denote f _ g.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Second...

Source: http://www.flooved.com/reader/1250

129
129

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 129

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Introduction: Before we try to solve higher order equations with discontinuous or impulsive input we need to think carefully about what happens to the solution at the point of discontinuity.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Second...

Source: http://www.flooved.com/reader/1401

147
147

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 147

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Separable Equations We will now learn our �rst technique for solving differential equation. An equation is called separable when you can use algebra to separate the two variables, so that each is completely on one side of the equation. We illustrate with some examples.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Nonlinear...

Source: http://www.flooved.com/reader/1426

113
113

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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eye 113

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First-order linear differential equations. We will give a systematic method of solving �rst-order differential equations (of normal form) y' + p(x)y = f(x) on a given interval I, where p, f are continuous functions.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1551

116
116

Nov 14, 2013
11/13

by
Daniel H. Rothman

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The question of stability is then addressed just as in the linear case, via calculation of the eigenvalues and eigenvectors.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1846

107
107

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 107

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Another important possibility which can in�uence how the trajectories look is if one of the trajectories traces out a closed curve C. If this happens, the associated solution x(t) will be geometrically realized by a point which goes round and round the curve C with a certain period T.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/1405

117
117

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 117

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We know that a standard way of testing whether a set of n n-vectors are linearly independent is to see if the n _ n determinant having them as its rows or columns is non-zero. This is also an important method when the n-vectors are solutions to a system; the determinant is given a special name. (Again, we will assume n = 2, but the de�nitions and results generalize to any n.)

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1454

105
105

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 105

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Remarks. Each of these three cases�one eigenvalue zero, pure imaginary eigenvalues, repeated real eigenvalue�has to be looked on as a borderline linear system: altering the coef�cients slightly can give it an entirely different geometric type, and in the �rst two cases, possibly alter its stability as well.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/1444

156
156

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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An equation for fundamental matrices: We have been saying �a� rather than �the� fundamental matrix since the system (1) doesn�t have a unique fundamental matrix: there are many ways to pick two independent solutions of x' = A x to form the columns of _.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1393

126
126

Nov 14, 2013
11/13

by
Hung Cheng

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Solutions expanded around an irregular singular point are distinctive in one aspect: they are usually in the form of an exponential function times a Frobenius series. Due to the factor of the exponential function, a solution near an irregular singular point behaves very differently from that near a regular singular point. It may blow up exponentially, or vanish exponentially, or oscillate wildly.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1338

116
116

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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eye 116

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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1388

117
117

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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We give a comprehensive development of the theory of linear differential equations with constant coef�cients. We use the operator calculus to deduce the existence and uniqueness. We presents techniques for �nding a complete solution of the inhomogeneous equation from solu-tions of the homogeneous equation. We also give qualitative results on asymptotic stability.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1562

180
180

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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eye 180

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The fundamental matrix _(t) also provides a very compact and ef�cient integral formula for a particular solution to the inhomogeneous equation x' = A(t)x + F(t). (presupposing of course that one can solve the homogeneous equation x' = A(t)x �rst to get _.) In this short note we give the formula (with proof!) and one example.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/1400

107
107

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 107

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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1450

102
102

Nov 14, 2013
11/13

by
Richard Melrose

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Remember Riesz representation theorem in Hilbert Space.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Methods,...

Source: http://www.flooved.com/reader/1483

148
148

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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We discussed some simple block diagrams when we introduced the notions of system, input, and output back in unit 1. Here, we will include the transfer function in the diagram and show how to use them to compute the transfer function of more complicated systems. As we do this, it will be useful to keep in mind the desciption of the transfer function as output/input.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Methods,...

Source: http://www.flooved.com/reader/1243

166
166

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 166

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In this section we will compute the unit impulse response as the limit of the responses to these box functions.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1392

97
97

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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As we have seen with the applet, Euler�s method is rarely exact. In this section we try to understand potential sources of error, and �nd ways to estimate or bound it.

Topics: Maths, Numerical Analysis, Error, Computation of Ordinary Differential Equations (ODEs) and Partial...

Source: http://www.flooved.com/reader/1256

207
207

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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eye 207

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The naive way to solve a linear system of ODE�s with constant coef�cients is by eliminating variables, so as to change it into a single higher order equation, in one dependent variable. One then solves this equation using the techniques for constant-coef�cient ODE�s learned in unit 2.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/1391

196
196

Nov 14, 2013
11/13

by
Gilbert Strang

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This section begins a major topic in scienti�c computing: Initial-value problems for partial di_erential equations.

Topics: Maths, Numerical Analysis, Error, Computation of Ordinary Differential Equations (ODEs) and Partial...

Source: http://www.flooved.com/reader/1686

125
125

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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eye 125

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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1434

102
102

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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An important result of mechanics is that a system of masses attached in (damped or undamped) springs is stable. A similar result is in network theory. In these notes, we study the differential equation of the form y" + py' + qy = f(t), where p, q are constants and f(t) represents the external forces.

Topics: Maths, Linear Algebra and Geometry, Differential Equations (ODEs & PDEs), Linear Algebra,...

Source: http://www.flooved.com/reader/1540

4,783
4.8K

Nov 14, 2013
11/13

by
Trench, William F., 1931-

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Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. In writing this book I have been guided by the these principles: 1. An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student�s place, and have chosen to err on the side of too much detail rather than not enough. 2. An...

Topics: Maths, Differential Equations (ODEs & PDEs)|, Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/3456

158
158

Nov 14, 2013
11/13

by
Hung Cheng

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In this lecture and in the next, we�ll briefly review second-order PDEs. We�ll begin with one of the simplest of such PDEs: the Laplace equation.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Methods,...

Source: http://www.flooved.com/reader/1148