PROFESSOR: Hi everyone.
Welcome back.
So today, we're going to take a look
at some forced oscillators and the exponential response
formula.
And the problem we're going to take a look at is first,
for part one, to consider the equation x dot dot plus 8x
equals, and then a forcing term on the right-hand side,
cosine omega*t.
And we're going to consider the case when omega squared is not
equal to 8.
So in the language of resonance, we're not on resonance.
And we're also asked why is this called
an undamped forced oscillator.
And then for part two, to use the ERF
to solve the differential equation x dot
dot plus 2x dot plus 4x equals cosine 3t.
And then, what is the natural angular frequency
of this differential equation?
So I'll let you take a look at these problems
and try them out for yourself.
And I'll be back in a moment.
Hi everyone.
Welcome back.
So we're asked to find the general solution
to the differential equation, x dot dot plus 8x equals cosine
omega*t.
And notice how this is a differential equation
with constant coefficients.
But it's being forced by a periodic function
on the right-hand side.
So the first thing to do is to write down
the homogeneous solution.
So the homogeneous solution, which
I'll denote with a subscript h, solves the differential
equation with the right-hand side of 0.
This gives us a characteristic polynomial of s
squared plus 8 equals 0, which then gives us roots
of plus or minus root 8 i.
So the homogeneous solution is some constant c_1.
And when we have purely imaginary roots,
we have cosine of the imaginary term times t,
plus some constant c_2 times sine of the imaginary term,
which in this case is root 8 t.
So the homogeneous solution is always contains two constants.
And it solves the differential equation
with a zero right-hand side.
The general solution, however, is the homogeneous solution
plus one particular solution that solves the differential
equation 8 dot dot plus 8x equals cosine omega*t.
So we just need to find one solution to this differential
equation.
And in this case, I'll use the exponential response formula.
But first note that when we use the exponential response
formula, we need a forcing to be of the form e to s*t
on the right-hand side.
And cosine omega*t is not in that form.
However, what we can do is there's
a trick to complexify the right-hand side.
So we note that the cosine omega*t is actually the real
part of i*omega*t.
So what this means is if we have a complex solution, zed,
which solves z dot dot plus 8z equals e to the i*omega*t,
then we can take x equals the real part of z,
then solves the differential equation x dot dot plus 8x
equals the real part of the right-hand side,
cosine omega*t.
And now we're in business because this equation
has the form where we can use the exponential response
formula.
So solving this differential equation for z,
we have a particular solution for z,
is going to be e to the i*omega*t divided
by the characteristic polynomial evaluated at the exponential.
So in this case, the exponential is i*omega.
So we must evaluate the polynomial at i*omega.
And this is the solution provided that the polynomial
evaluated at i*omega doesn't vanish.
And in our case, the characteristic polynomial,
p of s, we worked out already to be s squared plus 8.
So this gives us a particular solution for z,
which is 1 divided by i*omega squared plus 8
on the denominator.
We have e to i*omega on the numerator.
And of course, 1 over i*omega squared plus 8 becomes 1 over 8
minus omega squared.
And we know we don't have a problem
because omega, we were told in the problem,
is not equal to the square root of 8.
So in this case, we know that the denominator is not
going to vanish.
And now what we need to do is just take x,
for the particular solution, to be the real part of z,
which is going to be the real part-- I'll write it out--
8 minus omega squared cosine omega*t plus i sine omega*t,
which gives us 1 over 8 minus omega squared cosine omega*t.
So the total solution, the general solution,
is going to be the sum of the homogeneous solution
plus the particular solution.
And in our case, that's going to be c_1 cosine omega*t plus c_2
sine omega*t plus 1 over 8 minus omega squared cosine omega*t.
Sorry.
These should be root 8's.
So this is the homogeneous part.
And this is the particular solution.
And this is the general solution.
So we were asked, also, why this is sometimes called
an undamped forced oscillator.
Well, it's undamped, because in the differential equation,
there is no damping term.
There's no term proportional to x dot.
And then, secondly, it's forced because we have a forcing
on the right-hand side.
We have an input function, f, which in this case is cosine
omega*t which doesn't depend on x.
And that forces the differential equation.
Also note that the forcing term gives
rise to part of the solution which is directly
proportional to the forcing.
In fact, it has the same frequency
but a different amplitude.
So this concludes part A.
Now for part B, we're asked to use
the ERF to solve the differential
equation x dot dot plus 2x dot plus 4x equals cosine 3t.
And again, we can use the same trick.
The right-hand side isn't of the form e to the i*3t.
But what we can do is we can write cosine 3t as the real
part of e to the i*3t, and then solve the differential equation
z dot dot plus 2z dot plus 4z equals e to the i*3t,
and then take x equals the real part of z.
And in this case, we're only looking
for a particular solution, which we can compute using the ERF.
So it's 1 over the characteristic polynomial
evaluated at 3i times e to the i*3t.
And what's a characteristic polynomial
of this differential equation?
Well, p of s is going to be s squared plus 2s plus 4, which
means that p of 3i is going to be 3i squared plus 2 times 3i
plus 4, which gives us negative 9 plus 4-- is negative 5--
plus 6i.
And putting the pieces together, we end up getting that x is
equal to the real part of 1 over negative 5 plus 6i e
to the i*3t.
And we can expand out the numerator using Euler's formula
to get cosine 3t plus i sine 3t.
And when the dust settles, I got 1 over 61--
let me check my notes-- minus 5 cosine 3t plus 6 sine 3t.
So I'll let you work at this last step for yourself.
And then, lastly, what is the natural angular frequency
of the differential equation?
Well, this is just some notation.
We sometimes call the natural angular frequency
to be the square root of the term proportional to x.
So in this case, the term two here
comes in as a damping term.
The natural frequency, which is sometimes
written as omega naught squared, is always
equal to this term, which is 4.
So the natural frequency, the natural angular frequency,
omega naught, would be the square root
of 4, which would be 2.
So I just like to quickly recap.
We've taken a look at forced oscillators.
And we computed their solutions using the ERF.
And one common trend, particularly when
we have an oscillating input, a forcing term,
is to first change the oscillating term
into a complex exponential, then compute
a particular solution using the ERF formula,
and then take the real part of the ERF solution
to recover a real solution to the ODE.
That gives us the particular solution.
And then, in some cases, we also have
to add the homogeneous solution to it
to get the full general solution to the ODE.
So I'd like to conclude here.
And I'll see you next time.