Math 201 Fall 2011 Tutorial 6
Tuesday October 18
TAs: Points need to emphasize
•
Order of polynomial
•
If the guessed solution is a solution of the homogeneous equation, multiply it by
x
•
what to do if
g
(
x
)
is a product of polynomials,
sin
and
cos
, and
e
ax
•
what to do if
g
(
x
)
is a product of polynomials,
sin
and
cos
, and
e
ax
1. Solve
y
00

2
y
0
= 4
x
+ 2
.
•
The general solution
y
g
to
y
00

2
y
0
= 0
,
–
the auxiliary equation is
λ
2

2
λ
= 0
,
λ
1
= 0
,
λ
2
= 2
–
y
1
=
e
λ
1
t
=
e
0
x
= 1
,
y
2
=
e
λ
2
t
=
e
2
x
–
So,
y
g
=
c
1
y
1
+
c
2
y
2
=
c
1
+
c
2
e
2
x
.
•
A particular solution
y
p
:
–
because the right hand side is a polynomial, we guess for a polynomial solution
y
p
with order
n
. Note that the left hand side has an order
n

1
, the right hans side
has an order
1
, so
n

1 = 1
,
n
= 2
, that is, we guess for a quadratic solution,
y
p
=
ax
2
+
bx
+
c
,
–
y
00
p

2
y
00
p
= 2
a

2(2
ax
+
b
) =

4
ax
+ 2
a

2
b
= 4
x
+ 2
–
compare the coefficients of the polynomials on both sides,

4
a
= 4
,
2
a

2
b
= 2
, so,
a
=

1
,
b
=

1
,
c
is undetermined, a free parameter, pick
c
= 0
for simplification.
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 Winter '10
 STEACY
 Polynomials, yp

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