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Chapter 1 Problem Set

Page history last edited by sidjaggi 13 years, 5 months ago

ERG2011A Advanced Engineering Mathematics

(Syllabus A)

Problem Set 1

Kreyszig 9th edition - 1.1 - 1.4, 1.7

 

  1. (Solution verification) State the order and degree of the ODE. Verify that the given function is a solution (Formula and Formula are arbitrary constants).

                      Formula

 

     2.  (Geometric method) Kreyszig 9th Edition Problem Set 1.2, problems 4, 8, 12, 13

          Graph a direction field. In the field graph approximate solution curves through the given point or points Formula by hand.

                    Formula

          Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a           formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

                    Formula

 

     3.  (Solving Degree 1 ODEs) Kreyszig 9th Edition Problem Set 1.3, problems 4, 5, 7, 8, 9, 10, 12, 14, 17, 19.

          Find a general solution. Show the steps of derivation. Check your answer by substitution.

                    Formula

          Find the particular solution. Show the steps of derivation, beginning with the general solution.

                    Formula

 

     4.  (Modeling)

          (a)  Kreyszig 9th edition Problem Set 1.1, problem 19 = Kreyszig 8th edition Problem Set 1.1, problem 26

               (19) (Airplane takeoff) If an airplane has a run of 3 km, starts with a speed 6 m/sec, moves with constant acceleration, and makes the run in 1 min, with what speed does it take off?

          (b)  Kreyszig 9th Edition Section 1.3, examples 2, 4, 5.

          (c)  Kreyszig 9th Edition Problem Set 1.3, problems 25, 29 (Hint: use the forms of the solutions of examples 2 and 4 of Section 1.2).

               (25) (Radiocarbon Dating) If a fossilized tree is claimed to be 4000 year old, what should be its Formula content expressed as a percent of the ratio of  Formula to Formula in a living organism? (Assume that the half-life of the decay of C-14 to C-12 is 5730 years.)

               (29) (Law of cooling) A thermometer, reading Formula, is brought into a room whose temperature is Formula. Two minutes later the thermometer                     reading is Formula. How long will it take until the reading is practically Formula, say, Formula? First guess.

 

      5.  (Exact equations and integrating factors)

          (a)  Kreyszig 9th edition Problem Set 1.4, questions 2, 5, 7, 9, 12, 13.

                Test for exactness. if exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is                 given, determine the corresponding particular solution.

               Formula

 

          (b)  Formula(Hint: Consider an integrating factor of the form Formula)

 

     6.  (Existence and Uniqueness of solutions)

          Kreyszig 9th edition Problem Set 1.7, questions 2 and 6.

          Formula

 

     7.  (Picard Iteration)

          Kreyszig 9th edition Problem Set 1.7, questions 10(B), (C).

          Formula

 

 

Comments (4)

sidjaggi said

at 9:43 pm on Sep 19, 2010

I just got this email from a student

"To Sid,

Is the question (12) in Q.3 should be solved by substitution?
If so, what should be sub?

best regards,
HenryLai- engg2011 student"

Thanks for your question, Henry. First of all, please remember to post your questions in the appropriate comment box on the wiki -- this is good for two reasons (a) Not just I, but the tutor(s), and indeed everyone in the class, can respond, and so you might get an answer more quickly (in fact, Samuel asked me this same question earlier, after class). (b) More importantly, other students can also benefit from your question, and the posted answer.

As to your question...

Substitution is a good guess. Now since I don't want to give away the answer, I suggest you try using MAXIMA to get the answer, and then work backwards... looking at the answer, can you guess what substitution might work?
Try it tonight, and if you still can't get an answer, ask again tomorrow in the tutorial, or in the revision before the quiz.

terrylau910630@... said

at 1:31 am on Sep 22, 2010

u=y/x is the one of the suitable substitutions in this case.

terrylau910630@... said

at 1:29 am on Sep 22, 2010

u=y/x is the one of the suitable substitutions in this case.

terrylau910630@... said

at 1:32 am on Sep 22, 2010

pls ignore this comment/.

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