Intro to Differential Equations Math 20D Midterm 2 Winter 2019

Name:
PID:
TA:




Math 20D : Midterm 2




Turn o↵ and put away your cell phone.
No calculators or any other electronic devices are allowed during this exam.
You may use one page of notes, but no books or other assistance during this exam.
Read each question carefully, and answer each question completely.
Show all of your work; no credit will be given for unsupported answers.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
If any question is not clear, ask for clarification.
You have 50 minutes.




1 out of 10 points
2 out of 10 points
3 out of 10 points
4 out of 10 points
Total out of 40 points

, 1. (10 points) Consider the 2nd order di↵erential equation

only
2
preimes only y 00 + y = 0
,


need 2X's
(a) (4 points) Express the given 2nd order di↵erential equation as a matrix system.


, MAYa)
Define variances Gnd Xi -
Xa Pres:
X) =
y(t) Xi =
y(t)
=
Xz
ex
Xal y"(t) y
y'(t)
= =
= =
X2 =




x=
( i0)(x2)
-




(b) (6 points) Find a fundamental set of real valued solutions for the matrix system from
(a). 9 4
*
.




3 U1 v(t) + (wee)
find Xie
=
:
I char eg
det 1A-XI) = 0
Xi = git)i)
bi




(g)
a+



at
(ii) = = 0 -

W
eit(i)

eathinaceaa
_




det
(=x) =
0




(cost (
0
(x)) x) 1(1) isn(t)
=
- -
= +


X +1 0 =
icost + i not
x2 1 =




X 11i
=
[
cost
Real
+
iun(t)
complex

ilost-ant
complexReal
2 complex Eger
vct) + i (w(t)
X1 = x + i B Don't need x2
complex
.

real
(A XI) u
-

,
= 0

Xi = 1j
(cost) i(W
=
+




((ir) "(()
=
0
-




/ CostUn]
(
=

) =0 _




-
(a) (a) a)i)
m= = =