Non
‐
Homogenous ODE
W
ill b
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ki
t
h
ODES
f th
f
We will be looking at non
‐
homogenous ODES of the form:
)
(
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x
g
cy
by
ay
We know that we can solve:
Let y
h
be the general solution to the homogenous system above, and let y
p
be the
0
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particular solution to
. The our final general solution will
be y = y
h
+y
p
to the equation
)
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x
g
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)
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x
g
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Why?
Thi
if
fi d j
t
)
(
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x
g
cy
by
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This means if we can find just one
particular solution that makes it = g(x),
we will have our general solution
based on solving the linear
)
(
)'
(
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)'
(
p
h
p
h
p
h
y
y
c
y
y
b
y
y
a
)
(
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(
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p
h
p
h
p
h
y
y
c
y
y
b
y
y
a
homogonous systems we did before!
p
p
p
h
h
h
y
by
ay
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'
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)
(
0
x
g

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Undetermined Coefficients
Th
id
f
d t
i
d
ffi i
t i t
ith
bl
t
The idea of undetermined coefficients is to come up with a reasonable guess as to
what could be substituted as y and make it equal to g(x) on the other side. But
we do not know what the coefficient will be. Instead we write the coefficients
as constants then substitute them into the equation to make sure left side =
as constants, then substitute them into the equation to make sure left side =
right side.
B t
h t
ld b
bl
l ti
?
But what would be a reasonable solution?
We should never choose a candidate for our particular solution to be something in
our homogenous solution. (All solutions to the homogenous produce 0, so how
could it produce g(x)?)