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**Unformatted text preview: **Math 415.01 - Dr. Huseyin Coskun - Exam 2 Practice
Problems - 10230 Class (Vutha) February 14, 2011 The following list of problems is meant to serve as an extra resource for your preparations
for Exam 1. This list is in by no means exhaustive, nor is it meant to predict the
questions in the exam. Instead it can be used as a tool to assess how well prepared you may
be, and to highlight problems that you might face. Pay careful attention to your timing
and clarity of ideas — as this is something that can greatly aﬁect the outcome of your
exam. (3.1.22) Solve the initial value problem 41/” - y = 0, 31(0) = 1, y'(0) = -5
Find B for which all solutions approach zero as t -—> 00 (3.2.8) Find the largest interval in which the given initial value problem will certainly have a
unique solution. Do NOT ﬁnd the solution. (t - 1)?!” ~ 3ty’ + 4y = sin“), y(-2) = 2, y’(-2) = 1 (3.3.11) If the functions yl and y2 are linearly independent solutions to y” + p(t)y’ + q(t)y = 0,
prove that clyl and c2y2 are also linearly independent solutions provided c1 and c2 are
not zero. (3.4.20) Find the general solution and sketch a graph of the solution and describe its behavior for increasing t.
7r 77 3/"+y=°’ 9(3) =2, y'(§)=-4 1 (3.5.18) Consider the initial value problem 9y” + 12y’+4y = 0, y(0) = a > 0, y’(0) = —1 (a) Solve the initial value problem. (b) Find the critical value of a that separates solutions that become negative than
those that are always positive. (3.6.15) Find the solution of the given initial value problem 9" - 231’ + y = te‘ +4. 11(0) = 1. y’(0) = 1 (3.7.10) Find the general solution to the given differential equation 61 1+t2 y”-2y’+y= (3.8.7) ( Refer Text ) (3.9.9) If an undamped spring-mass system with a mass that weights 6 lb and a spring constant
1 lb/z'n is suddenly set in motion at t = 0 by an external force of 4 cos(7t) lb, determine
the position of the mass at any time and draw a graph of the displacement versus t. (10.1.15) Find the eigenvalues and eigenvectors of the given boundary value problem. 14” + M) = 0, y’(0) = 0, 9(a) = 0 ( Note: There are no problems from sections 10.2 and 10.3 ) ® Scuba“; m yCe):<:i?)€/Vg+(w)ézyb
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