ü Solution to Basic Example 1: Find a n, write series solution y HtL = n=0 a n ÿ t n and verify y(t) is a solution
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1 Basic Example 1: Follow the computations below to find the 4th order series approximation of the exact solution. Can you find/guess the 5th, 6th, nth order approximation? (Hint: For nth order approximation, look for pattern in a n by doing no simplications, see solution below)? ü Solution to Basic Example 1: Find a n, write series solution y HtL = a n ÿ t n and verify y(t) is a solution
2 2 SeriesNotesAndHomework.nb Basic Example 2: Follow the computations below to find the 4th order series approximation of the exact solution. Can you find/guess the 5th, 6th, nth order approximation? (Hint: For nth order approximation, look for pattern in a n by doing no simplications)? Use the nth order approximation to find the series solution y(t)? Can you verify that y(t) is a solution to the DE as done if Example 1? (see solutions below). ü Solution to Basic Example 2: Find a n and write series solution an ÿ t n
3 SeriesNotesAndHomework.nb 3 Example 3: Follow the computations below to find the 4th order series approximation of the exact solution. Can you find/guess the 5th, 6th, nth order approximation? (Hint: For nth order approximation, look for pattern in a n by doing no simplications)? Use the nth order approximation to find the series solution y(t)? Can you verify that y(t) is a solution to the DE as done if Example 1? (see solutions below). In[1236]:= ü Solution to Example 3: Find a n and write series solution y HtL = an ÿ t n
4 4 SeriesNotesAndHomework.nb Example 4: Follow the computations below to find the 4th order series approximation of the exact solution. Can you find/guess the 5th, 6th, nth order approximation? (Hint: For nth order approximation, look for pattern in a n by doing no simplications)? Use the nth order approximation to find the series solution y(t)? Can you verify that y(t) is a solution to the DE as done if Example 1? (see solutions below). In[1240]:= ü Solution to Example 4
5 SeriesNotesAndHomework.nb 5 Homework Problems Similar to Exs 1-4: Find series solutions y(t)= an ÿ t n to the IVPs (1) y -3y=0; y(0)=1 (2) y = y - 1; yh0l = 1 (3) y + y = t; y(0)=1 (4) y + t ÿ y = -y; yh0l = 1 (5) ẏ. + y = t; yh0l = 1, y H0L = 0
6 6 SeriesNotesAndHomework.nb Interesting Problem 5 (Gaussian Distribution): Show that y(t)= t 2 n g HtL = e -t2 is solution to the IVP. ü Partial Solution to Example 5 n! is a solution to y + 2 t ÿ y = 0; yh0l = 1. Show also that
7 SeriesNotesAndHomework.nb 7 Interesting Problem 6 (Schrodinger Equation): Verify that y(t)= 2 ÿ I -1 n! 2 Mn ÿ t 2 n+1 is a solution to the Schroedinger equation (the n=1 version) ẏ. + I3 - t 2 M y = 0; yh0l = 0, y H0L = 2. Show that y HtL = 2 t ÿ e -t2 ë2 is also a solution to this equation and that it is the same as the series solution. ü Partial Solution to Example 6
8 8 SeriesNotesAndHomework.nb Interesting Problem 7 (Bessel Functions): The Bessel function, denoted J 0 HtL is a special function (i.e. a function defined as the solution to a particular DE) given by J 0 HtL = H-1L n 2 2 n ÿhn!l 2 ÿ t2 n. Show that J 0 HtL is a special function for the DE (called Bessel s equation) t 2 ÿ ẏ. + t ÿ y + It 2 - n 2 M ÿ y = 0 with initial conditions yh0l = 1, y H0L = 0 when.
9 SeriesNotesAndHomework.nb 9 Interesting Problem 7 (Bessel Functions): The Bessel function, denoted J 0 HtL is a special function (i.e. a function defined as the solution to a particular DE) given by J 0 HtL = H-1L n 2 2 n ÿhn!l 2 ÿ t2 n. Show that J 0 HtL is a special function for the DE (called Bessel s equation) t 2 ÿ ẏ. + t ÿ y + It 2 - n 2 M ÿ y = 0 with initial conditions yh0l = 1, y H0L = 0 when.
10 10 SeriesNotesAndHomework.nb, ü Solution to Problem 7:
11 SeriesNotesAndHomework.nb 11 Reading and More Interesting Problems: Read SeriesAndSeriesSolutionsExcerpt.pdf and Try More of These Interesting Problems Below: H1L A form of Schroedinger' s Equation is "y''hxl+h2n+1-x 2 LÿyHxL =0". Find the series solution to Schroedinger' s equation when n = 0 and y H0L = 1, y' H0L = 0. H2L Can you find out the series expression of J 1 HtL H1 for n = 1L and show that it satisfies the appropriate Bessel equation? H3L Can you show that H4L For any and all of the problems above, can you determine the interval of convergence? H5L Here are some interesting new functions as explained on pg. 725 of SeriesAndSeriesSolutionsExcerpt.pdf. Can you solve them and research a particular function of interest to you to discover more about where they occur?
12 12 SeriesNotesAndHomework.nb,
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