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hey everyone and welcome to the second
video from chapter one format to 44 this
video is going to be about classifying
differential equations so this is going
to be a pretty terminology definition
heavy video but this is kind of the
basics that we need to be able to start
talking about differential equations the
class so we're going to talk about
different ways we classify them here and
go on talk about things in the rest of
this section so let's go ahead and jump
right into that.
so the first sort of
classification that we're going to start
out with is the difference between
ordinary differential equations partial
differential equations and systems of
differential equations so the first of
these we're going to talk about our
ordinary differential equations and
these are commonly abbreviated o des and
what this really means is that any folks
never looking for any of these functions
that we care about only depend on one
variable so first we can function y that
depends on X it only depends on X and
from that we can get a differential
equation say something it looks like
this and that gives us an eau de for
this function y it tells us how y in its
derivatives relate to each other and
since why only depends on one unknown
variable name one dependent variable X
it is an ordinary differential equation
the second type of equation that you'll
see our partial differential equations
and these are as you could probably
guess commonly abbreviated pdes and
these occur when you have a function
that you're looking for that depends on
more than one independent variable so
for instance I could say something like
the heat equation the wave equation both
of which are pds so in that sense we
have say a function you that depends on
t x and y and i can write a partial
differential equation that relates some
derivatives of you to each other for
instance the heat equation look
something like this as you can see in
this equation all the derivatives that
i've written our partial derivatives and
so we have a partial differential
equation where in the first case
alternatives that were ordinary
derivatives or normal just derivatives
in like a calc one sense and that we
call an ordinary differential equation
the last sort of thing that we use to
classify them are systems of
differential equations and the main
difference here between are things we
talked about earlier and systems are
that persistence we have more than one
unknown function so in the last case we
had our unknown function why are unknown
function you we only had one at a time /
systems you have multiple unknown
equations you're trying to solve for and
you have differential equations
equations on derivatives that relate all
of them to each other so one of these we
talked about in the first video was the
predator-prey model or a lockable terror
model which i'll rewrite here which is a
system of differential equations because
you have X derivative of X depending on
both x and y driven y depending on both
x and y and you want to solve these
together to get one solution that solves
both them at the same time so you're
trying to solve a system of differential
equations there so that gives the three
main classifications as a whole for
different situations you have your Orin
differential equations where your
function only depends on one variable
you have your partial differential
equations so ordinary depends on one
variable partial differential equations
where your function depends on more than
one variable and your derivatives are
now partial derivatives instead of just
ordinary derivatives and systems where
you have multiple equations or try to
solve for also functions trying to solve
for and a different set of equations to
try to solve those functions so that's
the first type of classification that
we're talking about today which is what
type of different situations do you have
our second type of classification we're
going to talk about is that the order of
differential equation and the order
differential equations is pretty easy to
determine in this case we're only
talking about order differentials here
we're not talking about any partial
derivatives systems can kind of go under
the same lines with mostly just a single
OD and the order of a differential
equation is just the highest number of
derivatives that appears somewhere in
the problem so all i care about four is
the biggest number that appears as a
derivative so for instance if i look at
the following oh de for a function x
where i have the fourth root of x plus 3
times the first derivative squared
equals 42 the fourth we see that the
highest order
derivative that shows up is the fourth
order derivative on the first term to
say this is a fourth order equation
right so this is not too hard to turn
meteors looking at the derivatives that
show up and what is the highest
derivative that appears anywhere in the
equation so that's our second form of
complication sort of talk about we have
whether it's an OD a PD or a system we
have what order the equation is and our
third and final we're talking about
today is whether or not an equation is
linear or non-linear so what I really
need for a quiz to be linear or
non-linear is looking at if it's linear
in terms of all of the unknown functions
so say we're looking at a PD in terms of
X so looking to solve for a function
that's going to be X of T now 24 the
equation to be linear I want that I
don't have any X terms multiplied by
each other and I don't have X's and
anywhere functions all I have is that
exits by itself or x a function of T so
for instance if I look at the following
oh de this OD looks pretty nasty but if
we look at all the X terms they all sit
by themselves I might have tea and some
weird function there's some nasty stuff
but the exes are all by themselves MX
third derivative plus some function in T
times X second derivative plus a
function T times the first derivative
plus the function G times X even the
function of T no X shows and weird
functions no x multiplied by yourself
itself so this is a linear equation and
click check what order is it it's a
third order equation now what are we
looking for for the equation is going to
be not linear so for instance look at
the following eau de now this OD is
nonlinear how do we know it's not linear
we know it's nonlinear because we see an
X prime squared term so that is not
linear in the X X prime variable its
squared so it's not linear there we also
have an e to the X both of those are bad
things we have X embedded in either x
squared function or an exponential
function both of which are not linear so
this is a non linear second order o de
and finally there are some other ways it
can be nonlinear so let's look at one of
those so this og we again have two
problems the first of which is the sine
term so
X being embedded in a sign is a bad
thing that's very nonlinear so that's
gonna make money to start we all set the
second term where we have X double prime
times X prime if their linear in each of
them respectively but since they're
multiplied together this is a nonlinear
equation because if i take two different
X's and put them together say I will end
up with a product going on here so if I
have a product of mixed terms even so
different derivatives it's still
nonlinear for that reason so this is a
nonlinear third order OD and those are
the three main types of classifications
that we have for differential equations
when they're not they're ordinary
partial systems whether what the order
of an equation is and whether their
linear or non-linear so that's it for
the material for this video at this
point I'm going to put up the video
question for this video so this is the
question you're going to want to answer
on your worksheet for this section and
give it to me in the next day of class
so i'll flip that up now alright so take
a look at that fill out the put that
onto your worksheet give all the answers
on that and then you can get to the next
day class alright thanks for watching
and I will see you all in the next video
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