webwork 3

①
'


Xy t 7
XYZ gli ) 7 day + PLX ) Q CX ) yn
=

y
= =

y
-




DX
'




y tyg
=

725
du
-
t ( I -
n )P (x )u = ( I -

n ) QLX)
a) Pix ) =L DX
X


Q (x ) =
7

n
= 2


' 2 '
b)
- -


=
u
y
=
y
' -
' '
u
y
=
y
-




off f f)
+ -
u =
-
7




""" " es #
" in
×=
¥
-




c) suit ) = e = =
e




uh ) f- HI
-
- '




=
7 In X t C
-




I
u x ) I
-


u
y
= =




ylx)

initial conditions :
X =
I
Ll ) 7
y
= -




Uli ) =
I =L
ya ) -
7



d) (x ) -7×1 nx
LyX
u
-
=




) y
'
(x )
-




-7×1 nx
IX
e = -




IX )
'


=/
Lx )
-




Tx in
y
-
x -

,②
¥y=y,÷ yet
de
-
-
I

dx


'
I #its
-




ya
①
3- -3=1×3
- " -
y dye y
-




DX




(
-
3
let u
y
=




-39¥ 54ft
-
-




"

EEF Fu Fs
- - -
-




,




II I Is
+ " = -




I.f .
=
ef¥d× =
×
6




uxb
f
'
DX
¥-3
= - .
X




=

f -
3×3 DX


= -
3×4 t C

4-

3
know that
-




u=y

3×6=-242×4 t C
-




y


initial
ya )
condition
= I




1 = -

I t C

4

I = C

4
'
13


⇐xii]
" "
'

, ③ 2yy
'
t 2 =

y
't 2x
y 101=6
'

Zyy
'
2x 2
y
-
= -




I
'
x I
yy
-
= -




2



'd
a) u =
y

un
y
-
-




I
'
-

y
2M


b) UM
'
u X I

21
=
. - -




÷ uz
x l
-
-
-_




'
U -
U
=
2X -
2



fp DX f -
DX - X

c) µc×)=e =
e =
e




=/
× ×
2)
-
-



e ulx ) e ( 2x -
DX


×

( ×dx
-




2)
-


= -


e ( 2x -
+ 2e


× ×
( 2x
- -


= -
e -
2) -

2e t C


×
U (X ) = -
( 2x -
2 ) -
2 + Ce



×
= -

2x tee



know that u=y2 and lo ) 6
-




y
-




2 ×
2x tee
y
= -




°

62 = -
210 ) t Ce



36 =
C


'
2x -136 ex
y
= -




't
=
l 2x -136 ex )
y
-