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PROFESSOR: Hi everyone.

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Welcome back.

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So today, we're
going to take a look

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at some forced oscillators
and the exponential response

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formula.

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And the problem we're going
to take a look at is first,

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for part one, to consider the
equation x dot dot plus 8x

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equals, and then a forcing
term on the right-hand side,

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cosine omega*t.

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And we're going to consider the
case when omega squared is not

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equal to 8.

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So in the language of resonance,
we're not on resonance.

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And we're also asked
why is this called

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an undamped forced oscillator.

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And then for part
two, to use the ERF

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to solve the differential
equation x dot

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dot plus 2x dot plus
4x equals cosine 3t.

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And then, what is the
natural angular frequency

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of this differential equation?

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So I'll let you take a
look at these problems

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and try them out for yourself.

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And I'll be back in a moment.

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Hi everyone.

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Welcome back.

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So we're asked to find
the general solution

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to the differential equation,
x dot dot plus 8x equals cosine

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omega*t.

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And notice how this is
a differential equation

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with constant coefficients.

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But it's being forced
by a periodic function

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on the right-hand side.

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So the first thing to
do is to write down

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the homogeneous solution.

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So the homogeneous
solution, which

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I'll denote with a subscript
h, solves the differential

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equation with the
right-hand side of 0.

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This gives us a
characteristic polynomial of s

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squared plus 8 equals 0,
which then gives us roots

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of plus or minus root 8 i.

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So the homogeneous solution
is some constant c_1.

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And when we have
purely imaginary roots,

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we have cosine of the
imaginary term times t,

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plus some constant c_2 times
sine of the imaginary term,

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which in this case is root 8 t.

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So the homogeneous solution is
always contains two constants.

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And it solves the
differential equation

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with a zero right-hand side.

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The general solution, however,
is the homogeneous solution

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plus one particular solution
that solves the differential

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equation 8 dot dot plus
8x equals cosine omega*t.

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So we just need to find one
solution to this differential

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equation.

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And in this case, I'll use the
exponential response formula.

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But first note that when we
use the exponential response

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formula, we need a forcing
to be of the form e to s*t

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on the right-hand side.

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And cosine omega*t
is not in that form.

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However, what we
can do is there's

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a trick to complexify
the right-hand side.

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So we note that the cosine
omega*t is actually the real

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part of i*omega*t.

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So what this means is if we
have a complex solution, zed,

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which solves z dot dot plus
8z equals e to the i*omega*t,

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then we can take x equals
the real part of z,

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then solves the differential
equation x dot dot plus 8x

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equals the real part
of the right-hand side,

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cosine omega*t.

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And now we're in business
because this equation

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has the form where we can
use the exponential response

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formula.

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So solving this
differential equation for z,

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we have a particular
solution for z,

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is going to be e to
the i*omega*t divided

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by the characteristic polynomial
evaluated at the exponential.

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So in this case, the
exponential is i*omega.

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So we must evaluate the
polynomial at i*omega.

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And this is the solution
provided that the polynomial

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evaluated at i*omega
doesn't vanish.

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And in our case, the
characteristic polynomial,

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p of s, we worked out already
to be s squared plus 8.

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So this gives us a
particular solution for z,

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which is 1 divided by
i*omega squared plus 8

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on the denominator.

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We have e to i*omega
on the numerator.

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And of course, 1 over i*omega
squared plus 8 becomes 1 over 8

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minus omega squared.

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And we know we
don't have a problem

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because omega, we were
told in the problem,

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is not equal to the
square root of 8.

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So in this case, we know
that the denominator is not

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going to vanish.

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And now what we need
to do is just take x,

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for the particular solution,
to be the real part of z,

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which is going to be the real
part-- I'll write it out--

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8 minus omega squared cosine
omega*t plus i sine omega*t,

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which gives us 1 over 8 minus
omega squared cosine omega*t.

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So the total solution,
the general solution,

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is going to be the sum of
the homogeneous solution

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plus the particular solution.

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And in our case, that's going to
be c_1 cosine omega*t plus c_2

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sine omega*t plus 1 over 8 minus
omega squared cosine omega*t.

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Sorry.

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These should be root 8's.

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So this is the homogeneous part.

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And this is the
particular solution.

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And this is the
general solution.

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So we were asked, also, why
this is sometimes called

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an undamped forced oscillator.

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Well, it's undamped, because
in the differential equation,

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there is no damping term.

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There's no term
proportional to x dot.

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And then, secondly, it's forced
because we have a forcing

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on the right-hand side.

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We have an input function, f,
which in this case is cosine

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omega*t which
doesn't depend on x.

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And that forces the
differential equation.

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Also note that the
forcing term gives

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rise to part of the
solution which is directly

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proportional to the forcing.

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In fact, it has
the same frequency

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but a different amplitude.

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So this concludes part A.

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Now for part B,
we're asked to use

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the ERF to solve
the differential

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equation x dot dot plus 2x
dot plus 4x equals cosine 3t.

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And again, we can
use the same trick.

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The right-hand side isn't
of the form e to the i*3t.

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But what we can do is we can
write cosine 3t as the real

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part of e to the i*3t, and then
solve the differential equation

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z dot dot plus 2z dot plus
4z equals e to the i*3t,

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and then take x equals
the real part of z.

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And in this case,
we're only looking

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for a particular solution, which
we can compute using the ERF.

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So it's 1 over the
characteristic polynomial

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evaluated at 3i
times e to the i*3t.

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And what's a
characteristic polynomial

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of this differential equation?

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Well, p of s is going to be s
squared plus 2s plus 4, which

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means that p of 3i is going to
be 3i squared plus 2 times 3i

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plus 4, which gives us negative
9 plus 4-- is negative 5--

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plus 6i.

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And putting the pieces together,
we end up getting that x is

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equal to the real part of
1 over negative 5 plus 6i e

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to the i*3t.

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And we can expand out the
numerator using Euler's formula

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to get cosine 3t plus i sine 3t.

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And when the dust
settles, I got 1 over 61--

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let me check my notes-- minus
5 cosine 3t plus 6 sine 3t.

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So I'll let you work at
this last step for yourself.

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And then, lastly, what is
the natural angular frequency

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of the differential equation?

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Well, this is just
some notation.

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We sometimes call the
natural angular frequency

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to be the square root of
the term proportional to x.

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So in this case,
the term two here

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comes in as a damping term.

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The natural frequency,
which is sometimes

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written as omega naught
squared, is always

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equal to this term, which is 4.

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So the natural frequency, the
natural angular frequency,

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omega naught, would
be the square root

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of 4, which would be 2.

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So I just like to quickly recap.

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We've taken a look at
forced oscillators.

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And we computed their
solutions using the ERF.

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And one common trend,
particularly when

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we have an oscillating
input, a forcing term,

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is to first change
the oscillating term

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into a complex
exponential, then compute

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a particular solution
using the ERF formula,

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and then take the real
part of the ERF solution

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to recover a real
solution to the ODE.

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That gives us the
particular solution.

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And then, in some
cases, we also have

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to add the homogeneous
solution to it

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to get the full general
solution to the ODE.

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So I'd like to conclude here.

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And I'll see you next time.