math 310 a11 - MATH 310 Fall 2011 Differential Equations...

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Unformatted text preview: MATH 310 Fall 2011 Differential Equations Homework Exercises, Set 11 Final Exam on Friday, 16 December 2011 Course Information is on WebCT Textbook references are to: Boyce and DiPrima “ Elementary Differential Equations and Boundary Value Problems ” (9th edition). This is the last problem set of the semester; recall: the Final Exam is on Friday, 16 December, 7:00–10:00pm in the Images Theatre . There is of course no quiz on this material on the Laplace transform, but it is important for exam preparation, and you are strongly encouraged to work through the problems — this material mainly needs some familiarity obtained through practice . Solutions will be posted by Friday, 9 December. The types of exam problems on the Laplace transform will be similar to those of previous semesters (see the old exams). A table of useful Laplace transform facts will be provided on the final exam; this table is available on the WebCT main page, and you are strongly encouraged to become familiar with it for reference. Know the definition of the Laplace transform! Reading : • Sections 7.6; 6.1, 6.2, 6.3 (6.4, 6.5, 6.6) Problems to prepare for exam (no need to hand in): • Section 6.1, # 4, 5, 6, 10, 13, 15 Laplace transform; for #5(c), show that L{ t n } = ( n/s ) L{ t n- 1 } = ( n ( n- 1) /s 2 ) L{ t n- 2 } = ... and hence deduce the formula for L{ t n } . Problems #26–27 on the Gamma function are also recommended as particularly instructive. Suggestion: do additional questions 1 & 2 (at least) before proceeding to the Section 6.2 problems. • Section 6.2, # 2, 8, 9, 10; 12, 16, 18, 21, 23 Laplace transform for the solution of initial-value problems • Section 6.3, # — The only material from Section 6.3 that will directly be needed for our final exam is the last page, p.328, Theorem 6.3.2 and the discussion below, which concerns shifting in the s-variable; the problems corresponding to this material are actually in Section 6.2, such as #9, 10, 16 and 21 • Section 7.5, # 15 Find the solution of the system using the Laplace transform (For a nonhomogeneous linear system, solve e.g. Sec. 7.9 #1 with x (0) = using Laplace transforms) • For further understanding of the power of the Laplace transform in solving differential equations, especially those with discontinuous, periodic or impulsive forcing, read and try...
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math 310 a11 - MATH 310 Fall 2011 Differential Equations...

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