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18.03 Class 15, March 13, 2004
Operators: Exponential shift law
Undetermined coefficients
[1] Operators.
The ERF is based on the following calculation:
D e^{rt}
=
r e^{rt}
=
rI e^{rt}
so
D^n e^{rt}
=
r^n I e^{rt}
and
(a_n D^n + .
.. + a_0 I) e^{rt} = (a_n r^n + .
.. + a_0) e^{rt}
or
p(D) e^{rt}
=
p(r) e^{rt}
So to solve
p(D) x = A e^{rt} ,
try
x_p = B e^{rt} ;
p(D) (B e^{rt}) = B p(D) e^{rt} = B p(r) e^{rt}
so we should take B = A/p(r) :
x_p = e^{rt}/p(r) .
What if
p(r) = 0?
eg
x"  x = e^{t} .
(*)
The key to solving this problem is the behavior of D on products:
(d/dt) (xy)
=
x' y + x y'
In terms of operators:
D(vu)
=
v Du + u Dv
Especially:
D(e^{rt} u)
=
e^{rt} Du + u r e^{rt}
=
e^{rt} ( Du + ru )
=
e^{rt} ( D + rI ) u
Apply
D
again:
D^2 (e^{rt} u)
=
D( e^{rt} (D+rI)u )
=
e^{rt} (D+rI)(D+rI) u
=
e^{rt} (D+rI)^2 u
Use: let's try a variation of parameters approach to solving
(*):
Try for
x
=
e^{t} u
Then
D^2 x
=
e^{t} (DI)^2 u
1]
x
=
e^{t} I u

e^{t}
=
e^{t} ( (DI)^2  I ) u
so want
( (DI)^2  I ) u
=
1
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.
 Winter '08
 Staff

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