IVLE8: [MA1506] Some Puzzles of Chapter 1

Dear all,

Here are some updating information in this week. All the contents are for the Mid-term Test.

Errata:

I found three typing errors in my tutorial notes and ultimate solutions, please have a look at this page (this page is updating when any error is found):

http://jin.chenyuan.googlepages.com/ma1506errata.

Students' Questions:

1) Some students can not understand the rule, used to find the particular solution to nonhomogeneous equations, which I mentioned in the tutorial and in tutorial notes , please have a look at the following file:

http://www.mediafire.com/?z4rziyvi3qw,
together with my tutorial notes 3:
http://www.mediafire.com/?3zbjdcljy2m.

Please remember: This rule can only applied to the equation with constant coefficients, and the right hand side is of the form:

(a) Polynomial
(b) Polynomial*e^(mx)
(c) Polynomial*sin(mx) or Polynomial*cos(mx)
(d) any linear combination of (a), (b) and (c)



For other cases,

(i) If the right hand side is not of the form (a) (b) (c) or (d), then first find the general solution y_h with right hand side being 0, then using the method of variation of parameters.

(ii) If the coefficients are not constant, we do not have effective methods to solve all the equations of variable coefficients, but for some special form we can solve it (for example Lecture notes example 25), you can find the solution here: http://www.mediafire.com/?8zwtvdgdggg.
(If you find a particular solution to the homogeneous one, you can use the method to reduce the order of the equation, and finally solve it.)

If these questions appear in the exam, I guess, the general solution to the homogeneous one will be given, then you need to use the method of variation of parameters to solve it!

(iii) If either the right hand side is not of the form (a) (b) (c) or (d), or the coefficients are not constant, then use the (i) and (ii).


2) How to solve the equation xy' = y - sqrt( x^2 + y^2 ), I mentioned a tip in tutorial notes 4.4.1, which is very important, and can be applied to this question:

http://www.mediafire.com/?unjd9tjtflb.

3) As Dr. Toh PC mentioned in his lecture, you can use Abel's identity to verify Wronskian, you can find details from wikipedia.org:

http://en.wikipedia.org/wiki/Abel_differential_equation.

4) More questions may be put into the web folder, if possible, please check it sometime later:

http://www.mediafire.com/?sharekey=cc44311791b417eb4012e8015643d9c82e81e8ebe912de5e.



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With best wishes to you test!

Chenyuan