MATH
80:
Homework Archive
This is an archive of the homework from weeks 1-3 and
the first midterm.
Week
1:
Learning
objectives
for week 1 (3/11-3/15). For this week you should:
- Review sequences, series and convergence.
- Review Taylor series and be able to compute a Taylor
series of a function
- Be able to define an Initial Value Problem (IVP).
- Understand Picard iteration and know how to use it to
generate a series of approximations to the solution of an
IVP.
- Understand the proof of the Existence and Uniqueness
Theorem for ODEs and know how to apply it to identify a
region where a given IVP has a unique solution.
- Here are some notes on Sequences,
Series and Convergence for you to review.
- Here are some notes on Picard
Iteration.
Homework 1 is due in class on Wed.
3/27
(Wednesday after spring break):
MAPLE
Solutions (to be posted on 3/27):
Week
2:
No
class
on Friday (3/29) due to the Cesar Chavez Holiday.
Learning objectives for
week 2 (3/25-3/27). For this week you should be able
to:
- Identify necessary conditions for an IVP to have a
convergent power series solution.
- Use Taylor series and repeated differentiation to find the
first few terms of a power series solution.
- Derive a recursion formula for the coefficients of a power
series for first- and second-order DEs.
- Generate the first few terms of a power series via a
recursion formula and be able to solve for all terms in some
instances.
- Identify when a power series has polynomial solutions
andsolve for the polynomials.
- Here are some notes on Power
Series for this week's lectures.
Homework 2 is due in class on Wed.
4/3:
Solutions (to be posted on 4/3):
Week
3:
Learning
objectives
for week 3 (4/1-4/5) and Monday 4/8. For these lectures
you should be able to:
- Given a second-order linear homogeneous DE, identify when
a point is an ordinary point, a regular singular point and
an irregular singular point.
- Identify when a DE is an Euler Equidimensional Equation.
- Solve Euler Equidimensional Equations for three cases
(real roots, repeated roots and complex roots).
- Know when it is appropriate to seek a Frobenius series
solution for a DE.
- Find indicial equations for a Frobenius series.
- Generate the first few terms of a Frobenius series via a
recursion formula and be able to solve for all terms in some
instances.
- Identify Bessel's Equation.
- Be able to identify Bessel functions of the first and
second kind, Jn(x) and Yn(x) .
- Find a Frobenius series solution for Bessel functions and
describe their behavior for small x.
- Describe a physical situation where Bessel functions
arise.
Midterm:
Due Monday 4/15 in class
The midterm was available on
Friday, April 12th and was due in class on Monday,
April 15th.
Average = 74.8 Median = 71 Standard Deviation
= 16.3
Here are the midterm
solutions.
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