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Includes study strategy for using prior review sheets and answer keys, plus core topics such as matrix exponentials eAt, determinants and inverses for 2x2 matrices, and solving constant-coefficient matrix differential equations via Laplace-based algorithms. Provides extensive practice problems on differential equations, Laplace transforms, inverse transforms, matrix systems, and a modeled pollution-inflow scenario.",{"@graph":69,"@context":123},[70,84,106],{"@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/math-2065-section-3-final-exam-review-sheet-comprehensive-laplace-transforms-and-matrix-exponentials/393964/",{"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/math-2065-section-3-final-exam-review-sheet-comprehensive-laplace-transforms-and-matrix-exponentials/393964.png","ImageObject",300,407,{"name":92,"@type":93},"Putri","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-09","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the final exam date, time, and location details?","Question",{"text":113,"@type":114},"The final exam is on Saturday, May 10 from 7:30 to 9:30 A.M. in the normal classroom.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Is the final exam closed book, and what reference is allowed?",{"text":118,"@type":114},"The exam is closed book, but the usual table of Laplace transforms will be provided.",{"name":120,"@type":111,"acceptedAnswer":121},"What study approach is recommended using the review sheets and answers?",{"text":122,"@type":114},"Complete the review sheets and exams without looking at answers, then compare with answer sheets to identify areas needing additional work. Additional review exercises of the same type are included here.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},393964,1791273984,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"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":46,"preview_page_count":4},962085571259,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Math 2065 Section 3  \nFinal Exam Review Sheet  \nThe ¯nal exam will be on Saturday, May 10 from 7:30 to 9:30 A.M. in the normal classroom. The exam is closed book, but you will be provided with the usual table of Laplace transforms. The ¯nal exam is comprehensive, and thus any of the material we covered is a valid source for questions.  \nYou should collect each of the review sheets and exams as sources for study for the ¯nal exam. Each of these is posted on the class web site, in case you have misplaced them. A good strategy for study is to do the review sheets and exams without looking at the answers. If you then compare with the answer sheets, you can identify the areas in which you need additional work. Some additional review exercises are included here, of exactly the same type as found in your text and on the previous review sheets.  \nFollowing is a brief list of terms, skills, and formulas covered after Exam 3 with which you should be familiar. Of course, you should also know the terms, skill, and formulas covered prior to Exam 3 .  \n² For a constant coe±cient homogeneous linear system  \ny0 = Ay ; y (0) = y 0 ;  \nthe unique solution is  \ny (t) = eAt y 0 ;  \nwhere the matrix exponential eAt is de¯ned by the in¯nite series  \n1  1   1   \neAt = In + At + 2 A2t2 + 3! A3t3 + ¢ ¢ ¢ + n!Ant n + ¢ ¢ ¢ :  \n(See Page 458 .) Know how to compute eAt from the de¯nition for some simple 2 £ 2 matrices, as is done in the examples on Pages 458-459 .  \n² You should know how to do the algebraic operations on 2 £ 2 matrices. Addition, scalar multiplication, and multiplication of matrices is reviewed in Section 9.1 . The following two facts for 2 £ 2 matrices are fundamental. If A = ·ac bd¸ then  \ndetA = ad ¡ bd  0 () A is invertible  \nand  \nif det A  0, then A¡ 1 = ad~~ ~~1¡~~ ~~bd · d¡c ¡ab¸ :  \n² If A = ·ac bd¸ , then the system of di®erential equations  \ny0 = Ay ; y (0) = ·c 1c 2¸ ;  \nwhich can also be written as  \nyy0102  acyy11 by 2dy 2 y 1 (0) = c1 ; y2 (0) = c2 ;  \nhas the solution  \ny (t) = eAt y (0) :  \nSince  \neAt = L¡ 1 © (sI ¡ A)¡ 1 ª ;  \n(see Theorem 6, Page 485) the following algorithm can be used to solve y0 = Ay when the matrix A is a constant matrix.  \nAlgorithm 1 . 1. Form the matrix sI ¡ A = ·s ¡¡ca s ¡¡bd¸ .  \n2. Compute the characteristic polynomial  \np (s) = det(sI ¡ A) = det ·s ¡¡ca s ¡¡bd¸ = s2 ¡ (a + d)s + (ad ¡ bd):  \n3. Compute  \n(sI ¡ A)¡ 1 = p~~ ~~1~~(~~s~~)~~ ·s ¡c d s b¡ a¸ = 426 spp¡~~(~~sc(sd)~~) ~~ psp¡~~(~~bs(s~~)~~a)357 :  \n4. Compute  \nL¡ 1 © (sI ¡ A)¡ 1 ª = 624LL¡1¡1½½spp¡~~(~~sc(s~~)~~d)¾¾ LL¡¡11½½psp¡~~(~~bs(s~~)~~a) ¾¾357 = ·h 1h 3((tt)) h 2h 4((tt))¸ :  \n5. The solution y(t) is then  \ny (t) = eAt y (0) = ¡L¡ 1 © (sI ¡ A)¡ 1 ª¢ y (0) =  \n\n| h 1 (t)\u003Cbr>h3 (t) | h 2h 4((tt))¸ | c 1c 2¸ = |\n| --- | --- | --- |\n\nc 1h 1c 1h 3((tt))  c 2h 2c 2h 4((tt))¸ :  \nReview Exercises  \nSolve each of the following di®erential equations.  \n1. y0 = t ¡ 2y  \n2. y0 = t ¡ 4ty  \n4  \n3. y0 + y = t4 t  \n4. yy0 = (t ¡ 1)2  \n5. y0 = 1 + t + y2 + ty2  \n6. For the equation y0 (1 ¡ t) = y ,  \n(a) Find the general solution.  \n(b) Find the particular solution with y(2) = 1, and give its interval of existence.  \n7. Consider the initial value problem ty0 = et ¡ y , y(1) = e.  \n(a) Without solving the equation, give the domain of existence of the solution, as guaranteed by the existence and uniqueness theorem.  \n(b) Now solve the equation and see if your answer is indeed de¯ned on the interval you found in Part (a) .  \n8. y00 ¡ 3y0 + 2y = 0  \n9. y00 + 2y0 + 2y = 0  \n10. y00 + 4y0 + 4y = 0  \n11. y00 + 6y0 + 9y = 0  \n12. y00 ¡ 6y0 + 13y = 0  \n13. y00 + 16y = 0  \n14. y00 ¡ 2y0 ¡ 3y = 0, y(0) = 0, y0 (0) = 1  \n15. y00 + 6y0 + 13y = 0, y(0) = 1, y0 (0) = ¡1  \n16. y00 ¡ 2y0 ¡ y = 0  \n17. y00 + 2y0 ¡ 15y = 0  \n18. t2 y00 + 2ty0 ¡ 6y = 0  \n19. 3t2 y00 + 11ty0 ¡ 3y = 0  \n20. t2 y00 + 9ty0 + 17y = 0  \n21. t2 y00 ¡ 3ty0 + 4y = 0  \n22. y00 ¡ 2y0 + y = 3e2t  \n23. y00 + 2y0 + y = 2e¡t  \n24. y00 ¡ y0 ¡ 2y = ¡9e¡t  \n25. y00 ¡ 2y0 + y = ett5  \n26. y00 +","cbCaikjNsWjXieDe","https://ap.wps.com/l/cbCaikjNsWjXieDe","pdf",125033,"English","# Final exam information\n# Study strategy and where to find materials\n# Core formulas and concepts\n## Matrix exponential eAt and its series definition\n## 2x2 determinant and inverse\n## Solving y'(t)=Ay with Laplace method\n## Algorithm steps for computing eAt\n# Review exercises\n## Differential equation problems (initial value and homogeneous/nonhomogeneous)\n## Laplace transform problems\n## Inverse Laplace transform problems\n## Matrix and system differential equations\n## Word problem: ammonium nitrate in a pond","[{\"question\":\"What are the final exam date, time, and location details?\",\"answer\":\"The final exam is on Saturday, May 10 from 7:30 to 9:30 A.M. in the normal classroom.\"},{\"question\":\"Is the final exam closed book, and what reference is allowed?\",\"answer\":\"The exam is closed book, but the usual table of Laplace transforms will be provided.\"},{\"question\":\"What study approach is recommended using the review sheets and answers?\",\"answer\":\"Complete the review sheets and exams without looking at answers, then compare with answer sheets to identify areas needing additional work. Additional review exercises of the same type are included here.\"}]","Math 2065 Section 3 - Final Exam Review Sheet - Comprehensive Laplace Transforms and Matrix Exponentials | PDF",1790396873]