Math 246, Spring 2009, Professor Levermore
Numerical Methods
For many first order differential equations analytic methods are either difficult or
impossible to apply. If one is interested in understanding how a particular solution behaves
it is often easi
Closed book. One page of notes. I
(I 7.3 .
1. Use series around x = 0 to solve:
a) y - 4 y = O; b) XA2y + xy + (XAZ - 9/4)y = 0.
447 M 1.0 I 2 {0 4",]
at) 4a : {swat ; ZMMM'JW = 42 4W
Qm+/(Mrl\m;44m.m maa
'1
~ 'v 1/5 , : Hm a
WMM M) 71 MI 77'
Ho ,;1/m
r READ THE DIRECTIONS. YOU MUST SHOW ALL WORK ON AND USE METHODS
LEARNED IN CLASS OR FROM THE WORKSHEETS TO RECEIVE FULL CREDIT.
CALCULATORS ARE PERMITTED. g .
100 pt. total .,
1. (16 pts) Differentiate the following:
a. y = ln(2x2 + x)3 + (27"2
2.
2 I Show all work. No credit will be given for unsupported answers. Circle your
answers. No calculators allowed. Staple this sheet to the top of your Exam.
1. [14 pts] Find the eigenvalues and the corresponding eigenvectors of
g LI 1 0 D
A = 4- 1 0
3 6 2
3
1. (20 points). Find constants A, B, C and D so that the plane containing the point (2,3,5)
and spanned by 171 : (2, 1,0) and 32 : (1, 3, 4) has the equation
Ax + By + Oz : D.
The normal vector for this plane is
3 E
(2,1,0)><(1,3,4): 2 1 0
1 4
7
Thus, t
Instructions: Please enter your NAME, your ID NUMBER, the FORM DESIGNATION
LETTER and your CRN NUMBER on the opscan sheet. The index number should be
written in the upper righthand box labeled Course. Darken the appropriate circles below
the ID number and
This part is closed book. You may use 1 page of notes. Follow the Honor Code.
1. Let f(x) = x"2 for O < x < 3 and have period 3.
a) Sketch the graph of f for 6 < x < 6. r
b) Can f be represented by a power series near 0? by a trig series? Explain.
c) If