Homework 6, due October 17, 2007.

Due Wednesday, October 17, 2007.
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goodwine
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Homework 6, due October 17, 2007.

Post by goodwine »

  1. Assume a solution of the form
    • Image
    to find a series solution to
    • Image
    Plot the solution for the first four and then five terms. Based upon the plot, what seems to be the range of time for which the series solution with the first 4 terms is valid?
    Hint: you could use a power series for the homogeneous solution and then undetermined coefficients for the particular solution, but since the inhomogeneous term is t you might was well just include it when you determine the coefficients in the power series.
  2. Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
  3. Assume a solution of the form
    • Image
    to find a series solution to
    • Image
    Plot the solution for the first four and then five terms. Based upon the plot, what seems to be the range of time for which the series solution with the first 4 terms is valid?
  4. Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
  5. Assume a solution of the form
    • Image
    to find the first five terms in the series for the general solution to
    • Image
    Hint: expand the sine and exponential functions in their Taylor series.
  6. Assume a solution of the form
    • Image
    to find a series solution to
    • Image
    Plot the solution for the first four and then five terms. Based upon the plot, what seems to be the range of time for which the series solution with the first 4 terms is valid?
    Hint: you want to equate powers of (t-1), not t.
  7. Write a computer program that uses Euler's method to compute an approximate numerical solution for the equation in the previous problem. On the same plot, plot the numerical solution and the power series solutions.
  8. Extra credit: (5 points each) for each of the problems determine a general form for the series (with a summation sign and an index on the coefficients).
  9. Extra credit: (5 points each) if you did the previous extra credit for each one, use your answers there to compute the range of times for which the power series will converge.
  10. Extra credit: (5 points each) for the problems where initial conditions were specified, plot the series solutions for 25, 50, 75 and 100 terms and compare with the approximate numerical solutions.
Notes:
  1. I have not done the extra credit problems myself (yet). Sometimes they can be pretty hard to work out; other times, not so much. At this point I don't know which case it is for any of these problems.
  2. The usual rules for programs apply: you must use Fortran, C or C++ for the program to determine approximate numerical solutions using Euler's method. You may use Matlab for all the plots.
  3. If the problem asks you to plot four and then five terms in the series, if one of the coefficients happens to be zero so that you are plotting exactly the same thing, then you must keep including more terms until you plot the next higher non-zero term in the series. The point is to add a term to see how it changes, which should indicate the range in which the truncated series is a good approximation. This shouldn't be confused with the range of convergence of the whole (infinite) series.
Last edited by goodwine on Mon Oct 15, 2007 11:24 pm, edited 1 time in total.
Bill Goodwine, 376 Fitzpatrick
Joe

Homework Problem 6

Post by Joe »

Since this equation is third order, don't we need a third initial condition to describe it?
mbeaucla

problem 5

Post by mbeaucla »

Are there initial conditions or should we just leave the answer in terms of a0 and a1?
goodwine
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Re: problem 5

Post by goodwine »

mbeaucla wrote:Are there initial conditions or should we just leave the answer in terms of a0 and a1?
The problem is to find the general solution, so you want to have two constants left in the solution.
Bill Goodwine, 376 Fitzpatrick
goodwine
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Re: problem 5

Post by goodwine »

mbeaucla wrote:Are there initial conditions or should we just leave the answer in terms of a0 and a1?
Oops, also assume x''(1)=1.
Bill Goodwine, 376 Fitzpatrick
goodwine
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Read this!

Post by goodwine »

Two things:

1. In the previous email I made a mistake on the initial conditions for
problem 6. If you haven't started yet, what I want for the initial
conditions is:

x(1) = 1
x'(1) = 0
x''(1) = 1

2. Extensions: if you really, really need an extension beyond this week,
you may have an extension until the Tuesday of break. If you leave you
must mail the homework to me (address below) and it must be postmarked
by next Tuesday. I want to post the solutions to homework 6 during the
middle of break.

If you only really need an extension, you may have until 5:00pm on
Friday or whenever you leave, whichever is earlier. Please try to meet
this one since break is supposed to be, well, a break...
Bill Goodwine, 376 Fitzpatrick
ececconi

Post by ececconi »

When graphing the solution to #6 approximately using euler's method, the starting t value can not be 0 due to the 1/t and 1/t^2. I was wondering, what would be a good starting value for t? .01? .001? 1?
goodwine
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Post by goodwine »

ececconi wrote:When graphing the solution to #6 approximately using euler's method, the starting t value can not be 0 due to the 1/t and 1/t^2. I was wondering, what would be a good starting value for t? .01? .001? 1?
Start at t=1 since that's when the initial conditions are given.
Bill Goodwine, 376 Fitzpatrick
Kron

Post by Kron »

I am having trouble with the first problem because when the x(t) is multiplied by t^2 the values for a4 and a5 go to zero and subsequentley all of my other values are going to zero as well. Is there something that I would be doing wrong because all I did was find the derivatives and then substitute into the equation and then solve for the values multiplied by each power of t.
goodwine
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Post by goodwine »

Kron wrote:I am having trouble with the first problem because when the x(t) is multiplied by t^2 the values for a4 and a5 go to zero and subsequentley all of my other values are going to zero as well. Is there something that I would be doing wrong because all I did was find the derivatives and then substitute into the equation and then solve for the values multiplied by each power of t.
The last part of the last sentence indicates you are basically doing the correct procedure. Without any more information, however, I can't really even make an educated guess as to what you are doing wrong. IIRC, it is not the case that all the terms after a4 are zero.
Bill Goodwine, 376 Fitzpatrick
Kron

Post by Kron »

how do we go about expanding the sin and exponential function using the Taylor series
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