Good day students welcome to two-part one of my slope fields uhm installment in this
tutorial was going to be going over how to determine the equation of the differential
equation given a slope or direction field okay so the instructions are as follows determining
the differential equation that could be represented the given slope fields. For number one we
have the sum circular slope field right here in the question is a which of these differential
equation our results in this slope field okay right so we are not the whole idea behind
the slope field we have a differential equation unity so we differential equation that C component
gives you a family of curves that our party solutions of the differential equation so
this slope is basically tells is the slope of the tangent line at each point on the family
of curves that represents the solution to the differential equation the one after she
the slope field to particular solution we have to find the initial value okay right
so let's some see what how we can determining which of these equations can be represented
by the Psalm slope field so the quickest way to designate do this using methods of elimination
so for Syria going to do is take a look at the slopes the slope field these are all tangent
lines and we ask yourself cardiol identical in here all identical that tells you that
the slope field is independent of the X or Y values it's a constant slope field in that
case your derivative are your differential equation is going to be equal to the constants
right so if you look at this of the field use noted that the all the same the have the
same steepness so to speak of this of this differed and this is see the steepness is
decreasing and it hits zero and then assertion goes into the negative of area answer can
see that it's not constants right since it's not constants we know that's the why the X
equals a constant cannot be answer so the can eliminate option a option a would have
been to beef the entire slope field had identical so okay for steepness right now that would
then we down with eliminated one optionally down to four over going to do is we're going
to fix one variable and ask ourselves how the other variable impacts the steepness of
the slope field okay so let's some fix our X how will fix our why first isolate six are
why is user red one here in red right so of if I wanted you speak this region up here
and the take this entire region why equals to I'm going to fix my why and when asked
myself the following question with wife fixed and X very in dollars the steepness of the
line change view these as tangent line sections of tangent lines okay to the steepness of
this tangent line change in the answer is absolutely yes so since it's changing that
means that X is going to be present in the differential equation okay right let's do
of take a look at the other orientation okay so the other orientation the going to look
at Tom look at fixing the of X the sixty X at X equals positive to for example okay X
equals published to see what that looks like that when the job start six right here okay
now look at these these of the slope lines right here this tangent lines okay if we hold
X fixed right and we allow white to attain different values does the alteration of the
variability of the why Valley impacts the steepness of the line the answer is yes you
noted that the slope of each line is different right here is undefined here is negative and
of negative and here its positive decreasing the steepness so to speak okay so is increasingly
increasing gets undefined in then he gets a negative in the subsequent is increasing
again so of this means that the why is going to be present in the differential equation
okay so says we, after doing are two investigations we know that in the differential equation
why and X are present the must be present in the differential equation because the vote
impact the steepness of the line will going to do that our out alternative and ask ourselves
which of these ones of has year X or Y a long okay and the answer is option be option B
has only and X okay so this cannot be a possible solution because X and Y impact play a role
in this has only X this equation is telling us that of why doesn't zero only X three zero
but in our meeting investigation just now we found out that both variables play a role
okay right now we have three options that's easy for us to determining which one the answer
is so what we do is the examine not slope field and we can look at the extreme cases
for example this is an extreme case right here what is the slope of this line this as
an undefined tangent line okay have an undefined tangent line acts when why is zero so that
points that point right there is one, zero so when why zero we notice that we have the
vertical tangents final undefined slope so a look at all this to three candidates that
we have which one results in undefined outputs when why is zero if you look at option C and
D D's to differential equations never result in a vertical tangent because they do not
have any variable in the denominator okay so you can never be see cannot be be be can
clearly see that you the answer because it will plug in the points on this of slope right
here with an again negative one over zero which is undefined right in this is exactly
what the slope of an undefined line line looks like okay so you see that the answer for number
one is option be okay right now let's take a look at number two so you gonna do the same
process let's examine the slope fields first and see you there is a pattern that we can
determine now what do you notice about the steepness of all the lights the noted that
all the lines have exactly the same steepness order that tell you it tells you that the
slope field is independent of X or Y it is a constant slope field right so the question
is which variables are going to be present the answer is none X and Y is independent
of both senior to have a look for the option that has neither X no one it the differential
equation is equal to the constant this has X and Y cannot be this cannot be them there
goes the answer right to option C this is this is the only one that satisfies the slope
is right here cannot be disconnected that right so let's see that is the answer makes
sense this answer tells is that the slope of the tangent line will always have a steepness
of negative one at everywhere the entire slope field is that the case here absolutely every
single him slope tangent line have the slope of negative one okay fifty the extended extrapolated
glacier you... Here so I look at this line right here from here to here or the steepness
of that line you dropped one negative one you run one so you see the slope of this line
is negative one and that's the slope of all the tangent line in our slope field right
so answer is option C so what on front a highlight here the different ways you can find the answer
values once you to see that of the connection or it's the shortcuts that you can use to
solve to determine what the graph is without actually doing any calculations whatsoever
right us take a look at the next of question question number three right us take a look
at is the field let's see if we can do some elimination here right first thing we going
to do look at all this tangent lines to deal have identical steepness the answer is absolutely
nuts okay this is positive let's positive zero negative and more negative right so we
know that he cannot be equal to the constant so you can easily eliminates option letter
D okay now let's look at the of the impact that each variable has on the slope field
first so that a look at the second at the impact that why has to determine the impacts
of the impact the X has to the sorry impact the X has we are going to of the going to
hold wife fixed okay so that a do is draw horizontal line want to fix why to a value
of two okay it why has a value of two constantly to and X is allowed to attain multiple values
negative to negative one zero one two does the variation of X impact the steepness of
the tangent line answer is absolutely sold that tells is that X is present X plus the
present okay X was the present in the differential equation now let's take a look at of us take
a look at the why is why impact the steepness of the state the lines in our slope do here
is to use a blind so let's peek this X Valley so with six X at the value of positive one
now let's take a look at this of this of region right here when X is constantly negative positive
one and why is allowed to attain multiple values namely to make one zero negative one
eight two why is allowed six have multiple values does that impact the steepness of the
line the answer is no so what does this tell us it tells is that this is differentially
quick this the field is independent of why so that means that why is going to be absent
from the differential equation right so does looking at that that that that that pattern
alone the can eliminate answer is right here okay so of the state forty equation that have
X and Y or just why okay this has why can never be the answer is this is depend on this
is the penicillin why this has just why cannot be the answer this has X to the candidate
in this has X this is also candidates our answer is either option C or option letter
D so which one is the answer will this is an easy one to determine because the take
a look at these two differential equations this guarantees that the slope will always
be positive because you square and inputs okay in this will take you give you basically
the opposite of the inputs is less the input positive one is slope is going to be negative
are one right to so this is always positive of the we have slope lines here that have
a negative steepness tangent line ~negative steepness the answer is yes at all these lines
to the right here the have negative slopes so this cannot be the answer the answer is
option letter D right of two that's that's that right now let's move on to the last problem
problem number four right so for number four if you look at the slope field the note is
that the all different so we know that he cannot be constant so you can quickly eliminates
option letter D right let's see what this umm slope field is independent of of so let's
take a look at how why impacts these of slope field by sixteen hour X okay so I fixed X
to negative to of do the slope of the tangent lines changed the answer is yes. The change
right so that means that why is going to be present in the differential equations so why
hasn't present right now let's fix our I'll why a look at the impact that the very alts
variability of X has on the steepness of the line okay so I fixed why and negative one
in I like I lets X Mary does that impact the steepness of the of tangent lines to the solution
of our differential equation the answer is no so the all have the same stupid tells me
that this equation is into the slope field is independent of X tens X is going to be
absent from the differential equation okay right so any equation that has X and Y I just
X cannot be the so the differential equation okay so this cannot be in this cannot these
are answer is either see heartbeat okay so how do you know which one of is the correct
one this was easy option number three because we can use X for the fact that this is a square
and only has positive outputs so we can figure that out but this is not the case so in order
to determine our what the correct answer these we just a peek at point right so we can take
this point right here easy point one zero two to point one, I'm sorry zero, one so right
inputs zero, one of into these two differential equations which would give me if the slope
that's something like this this line looks like a line what the slope of of negative
one right so let's see that plug-in zero, one which is the slope of negative one so
for this test this equation first option C we have why prime is equal to negative one
over why so at of a two zero one is going to be negative one over one which is negative
one exactly perfect this is what I'm looking for so this is the correct answer is I tested
option be can put a checkmark here this is get back tested option DY will that happened
if I tested option be why prime equals one over why the right inputs my why values into
the differential equation our get one over one which is one which is the line that has
a positive slope both can see that the steepness of this tangent line right here is negative
so option key is not the answer to final answer is option C okay so that's that the thanks
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