Exam 3 Review Sheet Math 2070

Transcription

1 The syllabus for Exam 3 is Sections 3.6, 5.1 to 5.3, 5.5, 5.6, and 6.1 to 6.4. You should review the assigned exercises in these sections. Following is a brief list (not necessarily complete) of terms, skills, and formulas with which you should be familiar. Know how to set up the equation of motion of a spring system with dashpot: + + = () where is the mass, is the damping constant, is the spring constant, and () is the applied force. Know how to determine if the system is underdamped, critically damped, or overdamped. Know how to solve the Cauchy-Euler equation = 0 where,, and are real constants by means of the roots 1, 2 of the indicial equation () = ( 1) + + = 0. Know how to use the reduction of order method to find a second solution 2 () of a homogeneous equation 2 () + 1 () + 0 () = 0 assuming that one nonzero solution 1 () is known. The second solution is assumed to be of the form 2 () = () 1 (). It is substituted back into the homogenous equation, resulting in a second order differential equation in in which does not appear, so the differential equation becomes a first order linear differential equation in. Solve for and then integrate to get, and hence 2 = 1. If 1 (), 2 ()} is a fundamental solution set for the homogeneous equation + () + () = 0, know how to use the method of variation of parameters to find a particular solution of the non-homogeneous equation + () + () = (). In this method, it is not necessary for () to be an elementary function. Variation of Parameters: Find in the form = where 1 and 2 are unknown functions whose derivatives satisfy the following two equations: ( ) = = (). Solve the system ( ) for 1 and 2, and then integrate to find 1 and 2. Know what it means for a function to have a jump discontinuity and to be piecewise continuous. 1

2 Know how to piece together solutions on different intervals to produce a solution of one of the initial value problems + = (), ( 0 ) = 0, or + + = (), ( 0 ) = 0, ( 0 ) = 1, where () is a piecewise continuous function on an interval containing 0. Know what the unit step function (also called the Heaviside function) (h( )) and the on-off switches ( [, ) ) are: h( ) = h () = [, ) = 0 if 0 <, and, 1 if 1 if <, = h( ) h( ), 0 otherwise and know how to use these two functions to rewrite a piecewise continuous function in a manner which is convenient for computation of Laplace transforms. Know the second translation principle: and how to use it (particulary in the form): L ( )h( )} = () L ()h( )} = L ( + )} as a tool for calculating the Laplace transform of piecewise continuous functions. Know how to use the second translation principle to compute the inverse Laplace transform of functions of the form (). Know how to use the skills in the previous two items in conjunction with the formula for the Laplace transform of a derivative to solve differential equations of the form and + = (), (0) = = (), (0) = 0, (0) = 1, where () is a piecewise continuous forcing function. Know the formula for the Laplace transform of the Dirac delta function ( ): L ( )} =. Know how to use this formula to solve differential equations with impulsive forcing function: + = ( ), (0) = 0 and + + = ( ), (0) = 0, (0) = 1. 2

3 The following is a small set of exercises of types identical to those already assigned. 1. Solve each of the following Cauchy-Euler differential equations (a) = 0 (b) = 0 (c) 2 12 = 0 (d) = 0 (e) = 0 (f) = 0 2. Find a second solution of each equation using the method of reduction of order. One solution is given. (a) (1 + ) + = 0; 1 () = (b) (1 + 2) = 0; 1 () = 2 (c) ( 2 + 2) = 0; 1 () = cos 3. Find a solution of each of the following differential equations by using the method of variation of parameters. In each case, denotes a fundamental set of solutions of the associated homogeneous equation. (a) = 1 ; =, } (b) + 9 = 9 sec 3; = cos 3, sin 3} (c) = 2 ; = 2, 3} (d) = 1/2 2 ; = 2, 2} 4. Graph each of the following piecewise continuous functions. 2 if 0 < 1, (a) () = 3 2 if 1. (b) () = 2( 1) h( 1) (c) () = [0,2) 3[2, 4) + ( 4) 2 h( 4) 5. Compute the Laplace transform of each of the following functions. (a) () = 2 h( 2) (b) () = ( 1) 2 [2, 4) if 0 < 1, (c) () = if 1 < 2 2 if Compute the inverse Laplace transform of each of the following functions. (a) () =

4 (b) () = (1 + /2 1 ) Solve each of the following differential equations. (a) 2 if 0 < = (0) = if 2 (b) + 4 = ( + 1)h( 2), (0) = 0, (0) = 1. (c) + 4 = 4( 2), (0) = 1, (0) = A mass hung from a spring behaves according to the equation + + = 0, where is the mass, is the damping constant, and is the spring constant. Suppose that = 1 kg, = kg m, and = 2 N/m (N=Newton). Suppose the following initial conditions are imposed: (0) = 1 m, and (0) = 1 m/s 2. Determine the values of for which the system is (a) overdamped, (b) critically damped, (c) underdamped. 9. Find the unique solution of the initial value problem = 0, (0) = 3, (0) = 1. Is the equation under damped or over damped? Does () = 0 for some > 0? If so, find the first > 0 for which () = 0. Sketch the graph of your solution. 4

5 1. (a) = (b) = 3/2 ( ln ) (c) = (d) = 1 cos(4 ln ) + 2 sin(4 ln ) (e) = 1 3/ (f) = 2 ( ln ) 2. (a) 2 () = (b) 2 () = (c) 2 = sin 3. (a) () = ( ln ) (b) () = (ln cos 3 ) cos sin 3 (c) () = (1/5) 2 ln (d) () = (4/15) 5/2 2 ) Answers ( 2 5. (a) () = ( 2 (b) () = ) ( ) (c) () = 1 ( ) ( ) 6. (a) () = 1 2 ( 3( 2) + 3( 2)) h( 2) (b) () = ( cos sin 3) + ( ( /2) cos 3( /2) 2 3 sin 3( /2)) h( /2) 7. (a) () = 2 5 (1 ( 5 ) + ) ( 2) ( 2) h( 2) (b) () = 1 2 sin 2 + h( 2) ( ( 2) cos 2( 2) 1 2 sin 2( 2)) (c) () = 1 2 sin 2 cos 2 + 2h( 2) sin 2( 2) 8. (a) > 8, (b) = 8, (c) 0 < < 8 9. Solution. The characteristic polynomial is () = = (+1) 2 +4 which has roots 1±2. Thus, the general solution of the homogeneous equation is = 1 cos sin 2. We need to choose 1 and 2 to satisfy the initial conditions. Since = 1 cos sin cos 2 2 sin 2, this means that 1 and 2 must satisfy the system of equations 3 = (0) = 1 1 = (0) = Hence, 1 = 3 and 2 = 1, so that the solution of the initial value problem is = 3 cos 2 + sin 2. 5

6 To answer the rest of the question, note that the characteristic polynomial has a pair of complex conjugate roots. Hence, the equation represents an underdamped system. To find the first > 0 such that () = 0, note that () = 0 3 cos 2 + sin 2 = 0 tan 2 = for some integer for some integer. 2 The smallest such positive is obtained for = 0 and hence the first time that () = 0 for > 0 occurs for = 3 cos 2 + sin