second midterm examination

ÇANKAYA UNIVERSITY
Department of Mathematics
MATH 258 - Introduction
to Differential
Equations
SECOND MIDTERM EXAMINATION
05.12.2016
STUDENT
t
__
\
1«;
Question
NAME-SURNAME:
"",I
SIGNATURE:
INSTRUCTOR:
DURATION:
Grade
Out of
NUMBER:
110 minutes
IMPORTANT
1
20
2
25
3
25
4
30
Total
100
NOTES:
ı) Please make sure that you have written your student number and name above.
2) Check that the exam paper contains 4 problems.
3) Showall your work. No points will be given to correct answers without reasonable
work.
Question 1. Find the general solution of the differential equation
yili
r>~"("
'l. + '2.. ~
r'l.
=
"l. ')..
«'~~)+ '2... ~('~
\'~.4 ıt:.
'cl
(' of-
- '3.6
A.) ~
L
+ yıl + 27y' + 27y = o.
i
('
e»
-=- +
•
'S-f'\ L
'."
Question 2. Solvethe
ıvP
yıl _ 3y'
~ z:
d \~
v'
Ci
'2..)(
e,'\ +~ e
-
+ 2y = 2ex + 4;
+ 'L
2...~e.l(
C\ ~ ~ +'Lt1..e ~X _'Le. ~ -'1... X
~i
y(O) = O, y'(O) =
~ (. E))
e: x,
d i(
l»
ı.
-=- o
.zzz;»
J{
=:>
:::.
Question 3. Find the general solution of
.2
X
ıı.; ~
of
~i (D+_-\)
Ctt~~
~
_ ~ 'D*
d~ "J
x > O.
~.
~,te> ) d - 's e L.i /;, + b e?·-t
+s= ~ \
'j
?
dt
alt1. -.,
d2y
dy
dx2 -6x
+10y=3.T4+6x3,
dx
e vi
i
/
e 5 i:
b
jp(+) "'- Ae 'it-+ B.e ~
d~f ... Y Ae lf{ + ~ ~e 1-LCL./:. -
JLJe. _ 1& s-e ~f
1
-\-~g e
+
dfl-
/GAe~\
_
2...
'31$ e3+ _ 2-1;" A-e L(~ _ 2~ I\,.e ,,+- 'i- A<> A"" ~+-le /c\l,..e 'ıL
A e.tti---'2
\.-\
-Lf:::: -
jHi:=.
jet.}=:
':$
~
f1<l.3t ==
Lt i:
e. ,....3
::i-n ~ ~t ""
1.
'1 -e ı'{i- + 'ers+-
A ~ - ~/L..
'" 1t-
Q..
C, {1.
ı.,ı:;
~
:=:>
's
-+ Lı- e ., to _ .~
l(
e, x -+ ~ x - 2. x - ~ x
'3
e. t; ~
::z.
'>, €.- 3f-
7, eVi -le {; e
rs =ı..
- ~
sf-
'."
Question 4.
a) Given that Yı(x) = x is a solution of the following homogeneous equation
(x
2
d2y
+ ı) -2
dx
-
dy
2x-
da:
+ 2y = o
find a second linearely independent solution.
(",\.ü. (XıJl(+'LiJ-l)
xLX~~>(J-'I
-:).)(
LX/f;
rA y~
_~y&~o
~2.ıY-I
,
tJlf
t
tJ-
2..
=- o
i
).cl w ~ ıY-
~C>
x (x\o
.2...
~) +-
W :::..D
:=<:>
Jw -...• ve>
2...
-
X
J)<.
()('\.t)
X{N)
aw
--=w
(
~~
')('l.-\.. 1\
_~)
dx
J
Y\W
::..
~V\(/:ij
b) Use the method of variation of parameters to find a particular solution of
, L X\A)1. _
G (x\'~)
X~A
-
, (~\/O (L-~)
------
Uxl.._xL-t.A
~
- {, ()(~,{ )
::::_'X'L-tb
':-
c) Find a general solution of