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REVIEW INFORMATION AND PROBLEMS FOR THE SECOND EXAM. The second exam will cover: (a) qualitative phase plane methods for differential equations; text, sections 7.17.5; (b) Inner products; orthogonal and orthonormal bases; expansion in orthogonal bases; pr

REVIEW PROBLEMS FOR THE FIRST MIDTERM (s 2)es . s2 4s + 3 (b) Compute the Laplace transform of (t 5) + g (t), where g (t) = sin t for 0 t < 2, g (t) = 0, for t 2. (Note: it may be helpful to use sin( + ) = sin() cos( ) + sin( ) cos(). (c) Find the Laplace

Homework 3, part 2: Solutions and remarks on selected problems Greenberg, 4.3, 6(a). The equation is 2x2 y + xy + x4 y = 0. (We have multiplied through by an additional factor of x to simplify bookkeeping.) Substitution of an xr+n 0 in the equation leads

Homework 3: Solutions and remarks on selected problems Greenberg, 4.3. In the solutions to this problem I will sometimes cite the following fact, which you may take as given: a rational function (that is, the ratio of two polynomial functions), is analyti

Homework Solutions: Problem 7 e) in 4.2 The equation is xy 2y + xy = 0. The rst problem is to obtain the radius of convergence of the series solution about the point x0 = 1. To do this rst write the equation in standard form 2 y y + y = 0. x In the notati

Homework 3: Solutions and remarks on selected problems Greenberg 5.5, # 7(f ). Solve x x = f (t), 1 et , 0 t < 6, and f (t) = 0 otherwise. x(0) = 0 = x (0), where f (t) =
As a preliminary, we compute the Laplace transform of f . This can be done directly

Homework 1: Solutions and remarks on selected problems Greenberg 5.2, # 1. A function f has exponential order as t if there exist constants K 0, T 0, and a constant c, such that |f (t)| Kect for all t T . (1)
A straightforward way to show that a function

642:527
The Gram-Schmidt Algorithm
The context of this discussion is a space of vectors V with an inner product denoted v, w . The norm associated to the inner product is denoted by v and is dened by v = v, v . Orthogonality and orthonormality are dened r

Frobenius method example for Sept. 20 lecture Preliminary remarks; using the Gamma function. The Gamma function often appears in the solution to recursion equations for series solutions to o.d.e. The Gamma function (x) is dened for all reals numbers x exc

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Entrance Guide, Math 527 This document is a summary of prerequisites that we shall particularly rely on in this class. From rst-year calculus, we will heavily use 1. Improper integrals. You should know how improper integrals are dened and how to calculate

640:527: Solutions Text, 19.2; 5(b) The pde is the wave equation c2 yxx = ytt with 0 < x < L, t > 0 and boundary conditions y (0, t) = 0, yx (L, t) = 0. By separation of variables y (x, t) = X (x)T (t) is a solution if X + X = 0, X (0) = 0 and X (L) = 0,