DIFFERENTIAL EQUATIONS MATH 308
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Date Created: 10/21/15
Math 308 Differential Equations Sections 504 and 507 Fall 2008 Final Exam PRACTICE Final PRACTICE The Final Exam will be in the usual classroom Name On my honor as an Aggie l have neither given nor received unauthorized aid on this academic work7 Signature of student Read all of the following information before starting the exam Show your work You may use your notecards but no calculators or other notes are permitted You do not need to simplify arithmetic expressions For example 12 7 02 is okay but 2 3 is not Circle or otherwise indicate your nal answers This practice test has 17 problems This is longer than the length of the actual Final Exam Its about twice as long The actual nal will be about 9 problems The Final Exam covers sections 1114 2126 31 32 34 36 37 41 49 5154 7178 selections from 8183 and 9195 It covers everything you7ve learned The actual test will be approximately 25 exam 1 material 25 exam 2 material and 50 the new stuff I cant have every single type of problem on the nal but there will be a nice selection Upcoming O ice Hours Wednesday December 3 from 1130AM 1230 Friday December 5 from 1130AM 1230 Monday December 8 from 1130AM 1230 I can stay past 1230 if there are people asking questions If these times dont work make an appointment Question 1 Solve the differential equation when 7 lt t lt 5 y 6y By tant Question 2 Solve the integro differential equation yt Ot e yt 7 1d1 t3 Question 3 Find a continuous function y that solves the differential equation y t it7 y0 1 when 07 tlt2 ft 6t7 2lttlt5 t2 5ltt Question 4 Solve the symbolic initial value problem y0 y 0 0 y 6y 17y 6t i 4 Question 5 Verify that fx 00 n0 n mm satis es the differential equation y0 17 y72zy0 Question 6 Find a degree 3 Taylor polynomial centered at z 0 that approximates the solution to the initial value problem y0 37 y 0 77 y fly1L emy lnz 1 Question 7 Part a Verify that zt SEW Cost cos t satis es the system Part b Find the eigenvalues of the following matrix 1 Question 8 Find a general solution to the system Question 9 part a Two 10 L tanks call them A and B are connected with a pipe which moves salt water from A to B at a rate of 10 Lmin7 and a second pipe which moves salt water from B to A at a rate of 2 Lmin Tank A has water with a salt concentration of 1 kgL entering from outside at a rate of 8 Lmin tank B has salt water draining to the outside at a rate of 8 Lmin Let t be the mass in kg of salt in tank A at time t Let yt be the mass in kg of salt in tank B at time t Set up a system of differential equations modeling this system part b Rewrite the system as a matriX differential equation Question 10 True always true or False 1 x 1y emy y 32x27 and y0 07 y 0 2 has a unique solution on the interval 173 2 DD D 7 where D 3 D i 42D 5y 2D 5D i 4y where D dim 4 2y 11 siny 32x27 and y0 07 y 0 12 has a unique solution on the interval 7171 5 2y will sinxy 32x27 and y0 07 y 0 712 has a unique solution on the interval 710710 6 6m 7 6 and 6 7 6m are linearly independent 7 leh sinz is a possible Wronskian for solutions to some equation y py qy 0 with p and q continuous on the entire real line 8 leh 6 is a possible Wronskian for solutions to some equation y py qy 0 with p and q continuous on the interval 17 oo 9 y x2y137 y0 yo has a unique solution on 7171 10 y x2 siny7 y0 yo has a unique solution on 7171 11 ft 171000 1000 has exponential order a for some a 12 ft 1t2 l0 10 2 has exponential order a for some a lt13 m 1 f0 14 f0106t71etdt e 15 f003956t71etdt e 16 6t71gtk 6t e 2 17 0 is an ordinary point of g 31y tanzy 0 18 7T2 is an ordinary point of g Sfily tanzy 0 19 The radius of convergence for a series solution to 2y 31y tanzy 0 cen tered at z 0 is at least 13 20 The following vectors which are solutions to a differential equation are linearly in dependent 21 The following vectors which are solutions to a differential equation are linearly in dependent 1 1 1 i2 7 0 7 i2 0 1 2 22 The following differential equation has a unique solution Mt 46 it W i who l 23 The following differential equation has a unique solution on the interval 174 51 l 1 l 961 2131512 l v 3 l 3129 l Question 11 Solve the differential equation Implicit solutions are ne dy 7 6 y26w2 dx i y Question 12 Solve the initial value problem y1 2 dy 2 72 xdx y Question 13 part a Use the direction eld to approximate the value of y3 for the initial value problem yl 74 Show your workl dyiy 22 dx x x2 213 m 2 part b Find an explicit solution to the initial value problem Question 14 part a A spring with mass 17 damping coef cient 107 and stiffness coef cient 24 is acted upon by an external force of sint Let7s ignore units for simplicity Write a differential equation for this situation part b Solve the differential equation Question 15 Part a Find the equilibrium solutions to the following system m M73 W ytwt Part b Solve the system Question 16 Find the Laplace transform of ft tet sint Question 17 Suppose 11t is the velocity in ms of a running turkey which takes the following values The turkey starts at 0 In Use Euler7s method with two steps to approximate how far the turkey traveled from 0 seconds to 1 second Overview for Exam 2 Sections 41 54 0 4 1 MassSpring Systems mm W W Flttgt dampedundamped forced unforced 0 42 Linear ODE7s Existence uniqueness when continuityl 0 43 Fundamental Solutions Wronskian determinant linear independence 0 44 Reduction of Order MI MIMI vz satis es equation of order one less 0 45 Linear Constant Coef cient Homogeneous ODE Solutions of the form eni Special Case CauchyEuler form az2y Izzyz cyz 0 0 46 Real Complex Multiple roots 0 4 7 48 Undetermined Coef cient 0 49 Variation of Parameters u1IyiI U2Iy2x o 4 10 4 11 412 ln depth look at massspring systems 0 52 Phase Plane 2 or more dimensions Critical points of autonomous system dz E f1 17 y d d i f 2 I 7 y 0 53 Elimination Methods 0 54 Coupled Spring Mass Systems Derivation of Equations 531 F U a MATH 308 Midterm I Review Categories of Differential Equations ODEPDE LinearNonlinear Order Initial Value Problems Euler s method for 2 aw 24950 yo Recursive formulas are zn1 n l 7747 yn1 39 l hfm 9n First Order Differential Equations a Separable Equations c Exact Equations Special Integrating Factors are not required b Linear Equations in the standard form dig Pmy Qzy use the substitution 1 yl d B ll E ti ernou1 qua ion dm Linear Second Order Differential Equations a Linearity of the differential operator 0 C d Homogeneous linear equations with constant coefficients Wronskian of two functions Linear independence of two functions i The auxiliary equation has two distinct real roots ii The auxiliary equation has one repeated real root iii The auxiliary equation has complex roots e Cauchy Euler Equations azzy my 0y hz Try the solution form y f Superposition principle g Method of undetermined coefficients for nonhomogenous linear equations Table will be provided Basics of complex numbers