HW 9 - New Deadline - monday 2nd november by 2.00 pm

Your total for A, B, C will be out of 40. Your total for D, E, F will be out of 40. I will enter them separately in my database of marks. Therefore, effectively this is two homeworks set at the same time. However, this is compensated by the fact that there will NOT be a homework due the week of the midterm.

If you succesfully complete this homework,  you will be in good shape for the midterm.

I'm giving questions number from the 9th edition when the question is taken from the book, and writing them out. If what I write differs from the book version, go with the book version (I'll try not to make any typos of course). Some of the questions are not in the book to make life more interesting.

A) [Total 14 marks] Reduction of order

Use the method of reduction of order to find a second solution to the differential equation

i) Section 3.4 qn. 23:  t^2 y" -4t y' +6y = 0, t  > 0, y_1 (t) = t^2
[6 marks]

ii)Section 3.4 qn. 28:   (x-1) y" -xy' +y = 0, x>1,  y_1 (x) = exp(x)
[8 marks] Hint: at some point in this question I think you will have to deal with an integration technique we haven't reviewed yet - that is the technique of integration by substititution. Check your notes or Wiki it!
 
B) [Total 14 marks] A miscellany of questions about springs.

Consider the ODE   u" + b u' + 2u = 0 for the motion of a spring.

i) What is the quasi-frequency of the motion when b=2?
[2 marks]

ii) What is the natural frequency of the motion when b=0?
[2 marks]

iii) Consider the motion when b=2 AND consider the motion when b=8.

In each case,

a) Can  a solution return to equilibrium in finite time?
b) Must a solution return to equilibrium in finite time?
c) Can a solution cross the equilibrium position exactly once?
d) Does every solution cross the equilibrium position at most once?
e) Can a solution cross the equilibrium position infinitely often?
f) Must a solution cross the equilibrium position infinitely often?

For all of the above, we exclude from consideration the solution u(t) = 0 (which corresponds to solving the ODE with initial conditions u(0)=0, u'(0)=0). Your answers will follow just from your knowledge of what possible graphs of solutions look like in each case.
[6 marks]

iv) For what value of w do each of the following ODE display resonance?
[2 marks each]

a) u" +u = cos(wt)
b) 10u" + 2u = 13cos(wt)

C) [Total 10 marks] Find the general solution of

i) y^(4) - 4y^(3) + 4y^(2) = 0 [5 marks]  (section 4.2 qn 14.) [I originally set qn 15. here but qn 15. requires properties of complex numbers that we haven't discussed]

ii) y^(4) - 25y = 0  [5 marks]

Here y^(6) means  the sixth derivative of y.

[A,B,C: total 38 marks + 2 for presentation]


D) [Total 10 marks] Find the Laplace transform

i) Compute the Laplace transform of f(t) = t
[5 marks] (This is section 6.1 question 5a)

ii) Compute the Laplace transform of f(t) = t u_1 (t),  where u_1(t) is the unit step function as defined in section 6.3
[5 marks]

E) [Total 12 marks] Using the table in the book, find the inverse Laplace transform of

i) Section 6.2 qn. 1:   F(s) = 3/(s^2+4)
[5 marks]

ii) Section 6.2 qn. 5:    F(s) = (2s+2)/(s^2+2s+5)
[7 marks]

F) [Total 16 marks] Using Laplace transform to solve ODE

i) (based on section 6.2 qn. 11) Suppose y(t) is the solution of y" -y' -6y = 0, y(0)=1, y'(0)=-1

Using the technique of chapter 6, (ie. NOT using the method of undetermined coefficients), find the Laplace transform Y(s) of y(t)
[6 marks]

Now find y(t) using the table for inverse Laplace transforms in the textbook.
[4 marks]

ii) Suppose y(t) is the solution of y' +2 y = t, y(0) =4. By taking the Laplace transform of the ODE, find the Laplace transform Y(s) of y(t). (Do not attempt to solve the ODE using first order methods). I do NOT require you to find y(t).
[6 marks]

[D,E,F: total 38 marks + 2 for presentation]