Find the fifth degree Taylor polynomial of the solution to the differential equation
Since y(0)=1, a0=1. Similarly y'(0)=-1 implies that a1=-1.
Since y''=3y, we obtain
y''(0)=3y(0)=3,
and thus a2=3/2.
Differentiating y''=3y yields
y'''=3y',
in particular
y'''(0)=3y'(0)=-3,
so
a3=-3/3!=-1/2.
Since
y(4)=3 y'', we get
a4=9/4!=3/8.
One more time:
y(5) = 3y''', so
a5 = 3(- 3)/5! = - 3/40.
The Taylor approximation has the form
f (
t) = 1 -
t +
t2 -
t3 +
t4 -
t5.
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Helmut Knaust
1998-06-29
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