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BU | MATH 226
100 sample documents related to MATH 226
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BU MATH 226
MA 226 Eulers method: January 30, 2008 Eulers method is the most basic of all of the numerical algorithms that are used to approximate solutions to dierential equations. Lets start with an example to get an idea of how the method works. Example. Co -
BU MATH 226
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BU MATH 226
Staple here Name MA 226 Solutions to Homework Last ve digits of ID number Solutions to exercises assigned during the week ending Discussion section (circle one): M 121 Homework format: 1. All solution sets must have this cover sheet. A pdf le for -
BU MATH 226
MA 226 Eulers method: January 26, 2004 Eulers method is the most basic of all of the numerical algorithms that are used to approximate solutions to dierential equations. Lets start with an example to get an idea of how the method works. Example. Co -
BU MATH 226
MA 226 Method of the lucky guess February 9, 2004 Last class we discussed how various solutions of a given linear equation are related. For homogeneous linear dierential equations, the Linearity Principle says that solutions are multiples of one an -
BU MATH 226
MA 226 Linear systems March 1, 2006 Last class we started to discuss linear systems, that is, the ones that can be written in vector form as dY ax + by = , cx + dy dt where a, b, c, and d are constants. These constants are also referred to as the c -
BU MATH 226
MA 226 Forced equations March 31, 2004 For the last six weeks, all of our dierential equations have been autonomous. Now we turn to second-order equations that model systems that are subject to some type of external forcing. Here are two examples: -
BU MATH 226
MA 226 Eulers method: January 30, 2006 Eulers method is the most basic of all of the numerical algorithms that are used to approximate solutions to dierential equations. Lets start with an example to get an idea of how the method works. Example. Co -
BU MATH 226
MA 226 Recall the following example from last class. Example. Consider dY = dt and the two solutions Y1 (t) = e-t 1 0 and Y2 (t) = et 1 1 . -1 0 2 1 Y March 3, 2004 We can solve any initial-value problem for this differential equation using an appr -
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BU MATH 226
Tacoma Narrows Bridge In this project, we discuss a model of the Tacoma Narrows Bridge. This model is based on the observation that while stretched cables can be reasonably modeled as springs, contracted (or slack) cables do not exert a restoring for -
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BU MATH 226
MA 226 March 26, 2004 We apply what we have learned about linear systems to solve second-order homogeneous linear equations. Let\'s return to the guessing technique for second-order equations that we learned about a month ago and see how it relates -
BU MATH 226
MA 226 More on the Laplace transform Last class we dened the Laplace transform. Denition. The Laplace transform of the function y(t) is the function Y (s) = 0 April 14, 2004 y(t) est dt. This transform is an operator (a function on functions). It -
BU MATH 226
MA 226 March 28, 2003 We apply what we have learned about linear systems to solve second-order homogeneous linear equations. Lets begin with one of the questions on Wednesdays exam: Example. Consider the equation 2 d2 y dy + 9 + 4y = 0. dt2 dt 1. -
BU MATH 226
MA 226 Linear systems February 27, 2008 Last class we started to discuss linear systems, that is, the ones that can be written as dx = ax + by dt dy = cx + dy dt dY = dt a b c d or as Y where a, b, c, and d are constants. These constants are als -
BU MATH 226
MA 226 Forced equations April 3, 2006 For the last ve weeks, all of our dierential equations have been autonomous. Now we turn to second-order equations that model systems that are subject to some type of external forcing. Here are two examples: Ex -
BU MATH 226
MA 226 A little more about Eulers method February 25, 2004 Eulers method for systems is just as easy to program as Eulers method for equations. Once again heres how we can program it with a spreadsheet. A 0 B 2 C 0 D 0.5 E F G 0 1 2 3 4 5 6 7 8 9 -
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BU MATH 226
MA 226 Linear systems February 28, 2003 Linear systems and second-order linear equations are the most important systems we study in this course. What is a linear system with two dependent variables? What is a second-order linear equation? How do -
BU MATH 226
MA 226 Homogeneous linear second-order equations March 24, 2008 Last class we revisited the guessing technique (guess y(t) = et ) for equations of the form a d2 y dy + b + cy = 0 dt2 dt (see Section 2.3 and notes for February 20 and 22). Lets see -
BU MATH 226
MA 226 Here are two examples of forced second-order dierential equations: Nonlinear pendulum: m d2 d + b + k sin = F sin cos t 2 dt dt April 2, 2003 Linear mass-spring system: m d2 y dy + b + ky = F cos t. 2 dt dt Our success studying unforced -
BU MATH 226
MA 226 Two examples of systems February 13, 2004 Example. Recall the predator-prey systems we discussed briey at the start of the semester dR = aR bRF dt dF = cF + dRF. dt F R, F R F R, F t R F R, F t R 1 t MA 226 We can compute two soluti -
BU MATH 226
MA 226 Analytic Techniques: February 20, 2004 There are few analytic techniques that work for both linear and nonlinear systems. 1. You can always check to see if a given function is a solution (no wrong answers). For example, consider the initial- -
BU MATH 226
MA 226 Linear systemsa brief review A linear system (with constant coecients) can be written as March 14, 2005 dY = AY, dt where A is a square matrix of constants (the coecients). For us, A will be a 2 2 matrix. Using the Linearity Principle, we c -
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BU MATH 226
MA 226 Method of the lucky guess February 10, 2003 Last class we discussed how various solutions of a given linear equation are related. For homogeneous linear differential equations, the Linearity Principle says that solutions are multiples of one -
BU MATH 226
MA 226 A little unnished business General solution of a partially-decoupled system Example. Consider the previous system dx = 2y x dt dy = y. dt February 24, 2003 We can calculate the general solution using methods we learned for rst-order equatio -
BU MATH 226
MA 226 Method of the lucky guess February 14, 2005 Last class we discussed how various solutions of a given linear equation are related. For homogeneous linear dierential equations, the Linearity Principle says that solutions are multiples of one a -
BU MATH 226
MA 226 Linear systemsa brief review A linear system (with constant coecients) can be written as dY = AY, dt March 13, 2006 where A is a square matrix of constants (the coecients). For us, A will be a 2 2 matrix. Using the Linearity Principle, we c -
BU MATH 226
MA 226 Existence and Uniqueness Theory for Systems March 2, 2005 There is an existence and uniqueness theorem for systems just like the theorem for equations. Existence and Uniqueness Theorem. Let dY = F(t, Y) dt be a system of dierential equation -
BU MATH 226
MA 226 Linear systems March 4, 2005 Last class we started to discuss systems that are linear. They are the ones that can be written in vector form as dY ax + by = , cx + dy dt where a, b, c, and d are constants. They are also referred to as the coe -
BU MATH 226
MA 226 A brief review from last class To solve a linear system March 16, 2005 dY = AY dt where A is a 2 2 matrix, we use the Linearity Principle applied to two linearly independent solutions Y1 (t) and Y2 (t). The general solution is k1 Y1 (t) + k -
BU MATH 226
MA 226 March 3, 2003 The Linearity Principle lets us produce many solutions from a few. Recall the example we did last class: Example. Consider dY = dt and the two solutions Y1 (t) = et 0 and Y2 (t) = et et . 1 0 2 1 Y Any linear combination of Y1 -
BU MATH 226
MA 226 More on the guessing technique for the damped harmonic oscillator February 22, 2008 Example. Last class we applied a guessing technique to produce two solutions for the harmonic oscillator equation d2 y dy + 3 + 2y = 0. 2 dt dt Using the cha -
BU MATH 226
MA 226 Guessing technique for the damped harmonic oscillator There is a guessing technique for the damped harmonic oscillator m d2 y dy + b + ky = 0. 2 dt dt February 23, 2004 1 MA 226 Example. Consider the harmonic oscillator d2 y dy + 3 + 2y = 0 -
BU MATH 226
MA 226 Section A1 Exercises March 21, 2008 1. A 4 kg mass is suspended from a spring in a liquid that offers a resistance force whose magnitude is eight times the velocity of the mass. (a) Find all spring constants for which the mass does not oscill -
BU MATH 226
Staple here Name MA 226 Solutions to Homework Last ve digits of ID number Solutions to exercises assigned during the week ending Discussion section (circle one): M 121 M 23 M 34 T 3:304:30 T 4:305:30 Homework format: 1. Solutions must be submit -
BU MATH 226
Staple here Name MA 226 Solutions to Homework Last five digits of ID number Solutions to exercises assigned during the week ending Discussion section (circle one): M 121 M 23 M 34 T 2:303:30 T 3:304:30 Homework format: 1. Solutions must be subm -
BU MATH 226
MA 226 Exam #1 Wednesday, February 11 All sections in Chapter 1 except Section 7 Getting help: 1. All sections meet as usual before the exam. 2. Anna and I will hold our usual office hours, tutoring hours, etc. before the exam. See the course web p -
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
MA 226 Forced equations March 31, 2008 For the last five weeks, all of our differential equations have been autonomous. Now we turn to second-order equations that model systems that are subject to some type of external forcing. Here are two example -
BU MATH 226
MA 226 Linear systems February 27, 2009 Last class we started to discuss linear systems, that is, the ones that can be written as dx = ax + by dt dy = cx + dy dt dY = dt ax + by cx + dy or as where a, b, c, and d are constants. These constants ar -
BU MATH 226
MA 226 Linear systems February 27, 2004 Last class we started to discuss systems that are linear. They are the ones that can be written in terms of matrix multiplication dY = AY. dt Recall two examples that we have already discussed. Example 1. We -
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BU MATH 226
MA 226 Existence and Uniqueness Theory for Systems February 25, 2008 There is an existence and uniqueness theorem for systems just like the theorem for equations. Existence and Uniqueness Theorem. Let dY = F(t, Y) dt be a system of differential eq -
BU MATH 226
MA 226 More analytic techniques 2. General solution of a partially-decoupled system Example. Consider the previous system dx = 2y x dt dy = y. dt February 23, 2009 We can calculate the general solution using methods we learned for rst-order equati -
BU MATH 226
MA 226 A translation from engineering terminology to mathematical terminology forced response-any solution to the forced equation. steady-state response-behavior of the forced response over the long term. April 6, 2009 natural (or free) response-an -
BU MATH 226
MA 226 Simple mass-spring system: Last class we derived the equation m d2 y + ky = 0 dt2 February 17, 2004 using Hooke\'s law. Let\'s consider the special case where k = m. We get d2 y = -y, dt2 and we can guess some solutions to this equation: In w -
BU MATH 226
MA 226 More general comments January 25, 2008 At the end of last class, I gave a rough description of what the differential equation dy = f (t, y) dt is and what it means to solve an initial-value problem. I have four more general comments: 2. Be c -
BU MATH 226
MA 226 January 28, 2005 At the end of last class, I made two general comments about rst-order dierential equations dy = f (t, y) dt and their solutions. 1. I gave a rough description of what a dierential equation is and what it means to solve an in -
BU MATH 226
MA 226 Existence and Uniqueness Theory for Systems February 25, 2009 There is an existence and uniqueness theorem for systems just like the theorem for equations. Existence and Uniqueness Theorem. Let dY = F(t, Y) dt be a system of differential eq -
BU MATH 226
MA 226 Forced equations April 1, 2009 For the last ve weeks of the semester, all of our dierential equations have been autonomous. Now we turn to second-order equations that model systems that are subject to some type of external forcing. Here are -
BU MATH 226
MA 226 Section A1 Exercises March 27, 2009 1. A 4 kg mass is suspended from a spring in a liquid that oers a resistance force whose magnitude is eight times the velocity of the mass. (a) Find all spring constants for which the mass does not oscillat -
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BU MATH 226
MA 226 Euler\'s method for a system February 26, 2003 We can use the vector field for a system to produce numerical approximations for the solutions. Example. Consider the IVP dx = -y dt dy =x-y dt (x0 , y0 ) = (2, 0). The EulersMethodForSystems t -
BU MATH 226
MA 226 Two examples of systems February 18, 2005 Example. Recall the predator-prey systems we discussed briefly at the start of the semester dR = aR - bRF dt dF = -cF + dRF. dt F R, F R F R, F t R F R, F t R 1 t MA 226 We can compute two so -
BU MATH 226
MA 226 More on Laplace transforms Denition. The Laplace transform of the function y(t) is the function Y (s) = 0 April 14, 2008 y(t) est dt. This transform is an operator (a function on functions). It transforms the function y(t) into the functio -
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BU MATH 226
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BU MATH 226
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BU MATH 226
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BU MATH 226
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