Model Challenge: Car Response Model Idea: Develop a model for the driver of a car following a second car. Since a car's speed is controlled only by the accelerator and brake (de-accelerator), you'll only be able to control the speed by changing the value
Math 512 Solutions to Homework #6 1. For t + (1 - 2)x = 0, the solution is implicitly given by (x, t) = f (x - t(1 - 2(x, t). And we know that (x, t) = f () on the line x - (1 - 2f ()t = x before a shock forms. x x (a) Let (x, 0) = f (x) where x0 f (x) =
Math 512 Homework #5 Due Thursday March 17 1. For the general solution of the system: dx dy dz = = , yz -xz xy(x2 + y 2 ) we start with the first two terms and after eliminating the common z term, get dx/y = -dy/x. Assuming y = 0 (e.g. y > 0), this is equ
Math 512 Solutions to Homework #4 1. Consider the following linear system of 1st order ODEs: u v (a) Let A= 2 -2 -1 3 , q(t) = f (t) g(t) , and y(t) = u(t) v(t) . = 2u - 2v + f (t) = -u + 3v + g(t)
Then with this new notation, the system becomes y = Ay +
Math 512 Solutions to Homework #3 1. Let y (t) + p(t)y(t) = q(t) with y(0) = y0 . (a) If p(t) = a and q = 0 (a constant), then y(t) = y0 e-at . (b) Let p satisfy a p(t) b for all t 0 and take y0 > 0. Let y, ya and yb be the solutions of y + p(t)y = 0, ya
Math 512 Solutions to Homework #2 1. (a) Let b(t) be the body's temperature, m(t) the medium's temperature and k the coefficient of heat transfer, then by Newton's Law of Cooling b (t) = -k(b(t) - m(t) or b (t) = k(m(t) - b(t). (b) A dead body is found in
Math 512 Solutions to Homework #1 1. (a) For y = tet , we solve by rewriting as dy = tet dt to get y = tet - et + C.
(b) For ty + 2y = t2 - t + 1, y(1) = 1/2, we rewrite as y + 2/ty = t - 1 + 1/t and use the integrating factor t2 to get (t2 y) = t3 - t2 +
Techniques for Solving 1st Order ODE-IVPs: y = f (t, y), y(t0 ) = y0 1. Direct Integration: y = f (t).
y(t) = y0 +
f (s) ds.
2. Linear, Constant Coefficient, Homogeneous: y + ay = 0. y(t) = y0 e-a(t-t0 ) or y0 exp(-a(t - t0 ). 3. Linear, non-Homogene
Quick Guide to Numerically Solving Systems of ODEs In modelling and other endeavors it is common to express some relationship using a system of ordinary differential equations (ODEs). It is also common, that in trying to make a realistic relationship one
Introduction to Modeling
A Collection of Lists Definition: A model is a representation of reality; a mathematical model is a model which uses mathematical objects (like functions and equations) to represent reality.
Properties of Models: Purpose: a model'
Math 512 Homework #7 Due Tuesday, April 26 1. For each of the following 2nd order PDEs, determine domain(s) in the (x, y)-plane where the PDE is (a) hyperbolic, (b) parabolic and (c) elliptic. 2uxx + 4uxy + 3uyy - u = 0 yuxx - 2uxy + ex uyy + x2 ux - u =
Math 512 Homework #6 (part I) Due Thursday April 7 1. With an appropriate change of variable, we can transform the traffic flow problem into the following: t + (1 - 2)x = 0, (x, 0) = f (x),
where now is the 'normalized' density with 0 1. The solution of t
Math 512 Homework #5 Due Thursday March 17 1. Find the general solution of the system: dx dy dz = = . yz -xz xy(x2 + y 2 ) 2. Find the general solution of (a) ux = 0 (b) yux - xuy = 0.
3. Find the general integral and compute three different solutions for
Math 512 Homework #4 Due Thursday March 3 1. Consider the following linear system of 1st order ODEs: u v = 2u - 2v + f (t) = -u + 3v + g(t)
(a) Rewrite in the form y = Ay + q(t) where y and q are vectors and A is a 2 2 matrix. (b) Find the eigenvalues and
Math 512 Homework #3 Due Thursday Feb 17 1. This is all about the linear ODE: y (t) + p(t)y(t) = q(t) with y(0) = y0 . (a) Find the solution when p(t) = a and q = 0 (a constant). (b) Suppose p satisfies a p(t) b for all t 0. Assuming y0 > 0 put the soluti
Math 512 Homework #2 Due Thursday Feb 3 1. Newton's Law of Cooling (and Warming): The rate of change of the temperature of a body is proportional to the difference between the body's temperature and the temperature of the surrounding medium. (a) Write an
Math 512 Homework #1 Due Thursday Jan 20 1. Find the solution of the following ODEs: (5 pts. each) (a) y = tet (b) ty + 2y = t2 - t + 1, y(1) = 1/2 (c) y = sec y (d) y = (e) y =
y 2 -1 , yt y 2 -1 yt
y0 = 1
, y0 = 1
(f) (et sin y - 2y sin t) + (et cos y +
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Gravitational & Fundamental Forces