We're solving the second order homogeneous constant
coefficient equation, and we're considering now
the second case when the roots of the characteristic
polynomial are complex. We need to figure out what to do with these complex exponential functions. So, let's work through the
theoretical idea first in this video, and then in the next video, I work an example. So, here we have solve
the characteristic equation, and let's say we found
a complex root Lambda plus i Mu, so Lambda and Mu are real numbers, and also we found the
complex conjugates. So, r bar equals Lambda minus i Mu. So, now we have two complex
exponential functions. We have z, which is e to the rt, so that's going to be e to
the Lambda t times e to the i Mu t, and then we have z bar, which is e to the Lambda t times
e to the minus i Mu t. So, it doesn't quite help us to have two complex functions
and solutions of our differential equation because we're looking for two real functions. The coefficients of
the differential equation are real, the initial conditions are real, and we know from
the numerical method, the Euler method that
we should be able to construct a real solution to
the differential equation. So, how do we go from two complex functions as
solutions to two real functions? The key is to use the principle
of superposition. So, using z and z bar, can we add them together in such a way that we
end up with two real solutions? The answer to that is yes. If we look at z plus z bar, and say divide that by two, you'll know from
complex numbers that this is just the real part of the function z or the complex number
z is just the real part, and that is what we're
going to call x_1, our real function x_1. How do you get
the imaginary part then? The imaginary part is z minus z bar divided by 2i. That will give us
the imaginary part of z, and that, we can call x_2. So, we have the real part of z and the imaginary part of z for
our two real functions. So, what is the real part
of this function, e to the Lambda t
times e to the i Mu t? Remember, so, recall that e to the i Mu t is Euler's identity cosine Mu t plus i sine Mu t. So, the real part of e
to the Lambda t e to the i Mu t is just e to
the Lambda t times cosine Mu t. So, our x_1 of t is e to
the Lambda t times cosine Mu t. The imaginary part
of z is x_2 of t, which is e to the Lambda t, which is a real exponential function, times the imaginary part
of e to the i Mu t, which is sine Mu t. So, let me summarize what I've done. We're solving our homogeneous constant coefficient
differential equation. We found two roots of
the characteristic polynomial, but they turn out to be complex. So, the roots are
complex conjugates of each other, I call them here r and r bar, and then with r equals
Lambda plus i Mu. So, we have two complex exponential
functions as solutions, e to the Lambda t times e to the i Mu t and e to the Lambda t
times e to the minus i Mu t, which is just the complex conjugate
of the previous one. Then we apply the principle
of superposition. So, we apply the principle
of superposition by adding z and z bar
and dividing by two, and then we get the real part of z, which is a real function that
we can use as our solution. We've apply the principle
of superposition again by forming z
minus z bar over 2i, that's just the imaginary part of z, and that will be our second solution. So, what are these two solutions? X_1 of t is e to the Lambda t, Lambda is the real part of r, times cosine Mu t. Mu is
the imaginary part of r. X_2 of t equals e to the Lambda t
times sine Mu t. So, we have put the real part of r
in the exponential function, and the imaginary part
of r shows up in the frequency argument of cosine and the frequency
argument of sine. So, this is what to remember. The real part of r goes in
the exponential function. The imaginary part of r goes
in the cosine and the sine. We will use this then in the next
video to serve as an example. I'm Jeff Chasnov. Thanks for watching, and I'll see you in the next video.