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Includes tasks on identifying functions from isoclines, sketching solution curves and phase lines, estimating solution values, and applying Euler’s method. Further problems address solving first- and higher-order differential equations, complex-valued trajectories, LTI systems with damped dynamics, Fourier series for periodic functions, convolution/impulse response, and generalized derivatives of piecewise-defined signals.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/practice-exam-3h-laplace-transforms-and-differential-equations/496113/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/practice-exam-3h-laplace-transforms-and-differential-equations/496113.png","ImageObject",300,407,{"name":92,"@type":93},"Jake","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-10-02",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What information and tools are allowed on this exam?","Question",{"text":112,"@type":113},"Calculators and notes may not be used. A table of Laplace transforms and other information is provided at the end of the booklet.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are the problems structured and what is the scoring?",{"text":117,"@type":113},"Problems are numbered 1 through 10, and some span more than one page. Total points are 300, with point values marked on the left margin.",{"name":119,"@type":110,"acceptedAnswer":120},"Which mathematical topics are included in the problems?",{"text":121,"@type":113},"The exam includes differential equations and isoclines, solution sketching and phase line analysis, Euler’s method, higher-order ODEs, LTI system dynamics, Fourier series, impulse response/convolution, and generalized derivatives.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},496113,1791153591,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":36,"preview_page_count":4},962084928904,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","18.03 Practice Exam 3h  \n\n| Your Name\u003Cbr>Recitation Leader\u003Cbr>Recitation Time |  |\n| --- | --- |\n\nDo not turn this page until you are instructed to do so.  \nWrite your name, your recitation leader's name, and the time of your recitation. Show all your work on this exam booklet. No calculators or notes may be used, but there is a table of Laplace transforms and other information at the end of this exam booklet. Point values (out of a total of 300) are marked on the left margin. The problems are numbered 1 through 10, but some span more than one page.  \nRecitation Leaders:  \nZachary Abel Eric Baer Ben Elias Abhinav Kumar Yifeng Liu Homer Reid Ruipeng Shen  \n\n| 1 |  |\n| --- | --- |\n| 2 |  |\n| 3 |  |\n| 4 |  |\n| 5 |  |\n| 6 |  |\n| 7 |  |\n| 8 |  |\n| 9 |  |\n| 10 |  |\n|  |  |\n\n1. (a){(d) concern a certain di􀀋erential equation dydx = f(x; y) . The 􀀌gure below shows the isoclines for this di􀀋erential equation with slopes m = 􀀀2 ; 􀀀1 ; 0 ; 1 ; 2, from left to right. Other isoclines follow the same pattern.  \n[4] (a) The function f (x; y) is one of the following. Which?  \n(i) x3 (ii) x 􀀀 y 3 (iii) y 􀀀 x3 (iv) y 3 (v) y 3 􀀀 x (vi) x3 􀀀 y  \n[5] (b) On the diagram above, sketch the graph of the solution with y(0) = 0 .  \n[5] (c) For this solution, estimate the value y(1000) . Is your estimate too large or too small?  \n[6] (d) Decide whether each statement below is true or false, and explain your reason.  \n(i) Every solution has exactly one minimum.  \n(ii) No solution has a maximum.  \n(iii) Any two solutions are asymptotic to each other as t ! 1 .  \n[5] 1. (e) Sketch the phase line and graphs of some solutions ofthe autonomous equation dydt = y 2 􀀀 2y. Include all equilibria (and mark them as stable, unstable, or neither) and enough solutions so that every solution is a time-translate of one drawn.  \n[5] (f) Use Euler's method with step size ~~1~~2 to estimate the value y(1) of the solution of dydx = 1 + xy with y(0) = 1 .  \n[20] 2. (a) Find the general solution of the equation 2xdydx + y =px sin(x) .  \n[10] (b) Suppose we know that the equation y0 + p(x)y = q(x) has among its solutions the functionsex and 􀀀e􀀀x. What is the solution y(x) with y(0) = 0?  \n[10] 3. (a) Find the cube roots of 􀀀8, and express each in the form a + bi.  \n[10] (b) Find positive real numbers A, 􀀞, and ! such that 2 cos(3t) + 2 p3 sin(3t) = A cos(!t 􀀀 􀀞) .  \n[10] (c) Sketch the trajectory of the complex-valued function f (t) = e (ln(2)+2􀀙i)t. Mark the positions of f(􀀀1), f (0), and f (1) .  \nd3 x  \n[10] 4. (a) Find the general real solution of dt3 + 8x = 0 . [3(a) may be useful here.]  \nd3 x  \n[10] (b) Find a particular solution of ~~ ~~ + 8x = 8 + cos(t) .  \ndt3  \n[10] (c) Find a particular solution of  + 4x = 25 te􀀀t.  \n5. Consider the LTI di􀀋erential operator p (D) = D2 + 8D + 25I. This is an underdamped system.  \n[10] (a) Find the damped angular frequency (i.e. the pseudo-frequency of solutions of p(D)x = 0) .  \n[10] (b) Find the frequency of a sinusoidal input which produces the maximal amplitude gain.  \n[10] (c) Express the tangent of the phase lag (relative to a sinusoidal input of angular frequency !) as a function of !.  \n6. A certain periodic function f (t) has Fourier series b 1 sin(􀀙t) + b2 sin(2􀀙t) + b3 sin(3􀀙t) + 􀀁 􀀁 􀀁 , where bn = 1=n if n is twice an odd number, and bn = 0 otherwise.  \n[10] (a) Please express f (t) in terms of the standard squarewave sq (whose Fourier series is given atthe end of this exam booklet) .  \n[10] (b) Determine the Fourier series of the periodic solution of +!2 x = f(t) (for the same function f as in (a)), if a periodic solution exists.  \n[10] (c) For what positive values of ! are there no periodic solutions?  \n7. A certain LTI operator p(D) has unit impulse response w(t) = t2 e􀀀tu (t) .  \n[6] (a) Find the solution of p(D)x = e􀀀t with rest initial conditions.  \n[6] (b) What is the characteristic polynomial p(s)?  \n7. In (c){(e) we let f (t) = ~~1~~2t2 (u(t + 2) 􀀀 u(t 􀀀 2)) .  \n[6] (c) Sketch the graph of f(t) .  \n[6] (d) Sketch","cbCairQTmlDZ8Xp4","https://ap.wps.com/l/cbCairQTmlDZ8Xp4","pdf",129426,16,"English","# Instructions and Setup\n# Problems 1-10\n## Problem 1\n## Problem 2\n## Problem 3\n## Problem 4\n## Problem 5\n## Problem 6\n## Problem 7","[{\"question\":\"What information and tools are allowed on this exam?\",\"answer\":\"Calculators and notes may not be used. A table of Laplace transforms and other information is provided at the end of the booklet.\"},{\"question\":\"How are the problems structured and what is the scoring?\",\"answer\":\"Problems are numbered 1 through 10, and some span more than one page. Total points are 300, with point values marked on the left margin.\"},{\"question\":\"Which mathematical topics are included in the problems?\",\"answer\":\"The exam includes differential equations and isoclines, solution sketching and phase line analysis, Euler’s method, higher-order ODEs, LTI system dynamics, Fourier series, impulse response/convolution, and generalized derivatives.\"}]","Practice Exam 3h - Laplace Transforms and Differential Equations | PDF",1790927899]