let's solve this differential equation we have y prime equal to X to a third power times the
parenthesis with 1 minus y inside and we also know Y of 0 is equal to 3 we are going to use
this to help us stop the C data okay so first let's rewrite the Y prime is the Y the X we know
it's DX because we have this extra here already so the dependent variable must be X as well okay this
is equal to X to the third power times 1 minus y and now let's go ahead and do our usual business
let's multiply by the X on both sides so they cancel and we are going to divide 1 minus y on
both sides on the left hand side I will just put this down as 1 over 1 minus y ok we have 1 over
1 minus y dy on the left hand side and this is equal to X to the third power DX and now we can
just go ahead integrate integrate all the voids are together now or the X are together not right
okay integral of 1 over 1 minus y is going to be Ln absolute value of that 1 minus y like this and
then be sure you know the derivative of 1 minus y is negative 1 we had to divide it by negative
1 so it's the same as just putting a negative in the front and don't worry about to put down plus
zi1 whatsoever right here just make this equal to and we integrate x plus third power which
is going to give us X to the fourth power over 4 and on the right hand side we put down the plus c1
right here and we are done with the solving part right with all the integration part as well but
then Joe I plug in this initial condition right here or not where it's kind of up to you this one
is not bad okay and I would like to just isolate a Y first and then plug in the initial condition
later on after I get the why I saw that okay so to do so you know this is negative Ln so let's
go ahead multiply both sides by negative one so that negative negative will become positive and
we will have Ln absolute value of 1 minus y and this is equal to of course we distribute negative
1 inside this times that is going to be negative x to the fourth power over 4 then negative 1
times c1 yes yes negative C 1 but then C 1 is a constant negative C 1 it's also a constant let me
write this down with no negative C wants another constant then we just call this to be another one
that's a c2 okay this is how you do arithmetic with constants negative C 1 is just a constant
so I would like to put this does plus 2 so much fun with constants ok all right to carry off the
iron let's do a to this power and that's 2 e to the power so that this and that will cancel and
keep in mind on the right hand side both of them had powers here we have absolute value of 1 minus
y and this is equal to when we are adding powers it's the same as saying e to this power and then
we multiplied by e to that power let me put down this first e to the C 2 right here and we multiply
this by e to the negative x to the fourth power over 4 and the reason I put into the c2 down first
it's because C 2 is a constant is a constant a constant - another constant is just another
constant so we might as well just call this to be another concept of course we keep labeling will
just use C 3 ok now we have the absolute value of 1 minus y is equal to this is going to give
us C 3 and then we have that e to the negative x to the fourth power over 4 how can I get rid of
the absolute value well that's just scary of the absolute value and be sure you put on a plus minus
le right inside on the left hand side we will just have one minus y but then right here so I keep
the pathway or should I keep the negative okay we have parsley C Street or negative C Street doesn't
really matter C streets are constant past these thin streets are constant negative C Street is
also another constant we might as well just call the whole thing right here to be another constant
I don't know yet how about that's your C for this right here is just going to be sieved what but now
and we'll figure out what this constant later anyways right so let me just write this down
and then we have this e to the negative X to the fourth power over 4 and of course next we can just
subtract 1 on both sides and then we will have negative Y which is equal to C 4 times e to the
negative x to a fourth power over 4 and then minus 1 after that the minus 1 is 9 the exponent is just
a regular minus 1 next of course divide everything by negative 1 so that we will get passed divided
by itself and you see C 4 divided by negative 1 is C 5 so let me just write it down 5 negative C
4 is just another constant let me use C 5 for that and we have the e to the negative X to the fourth
power over 4 negative one over negative one yes I know it's just another concept but I know how
the content is is just parsley one so write down plus one we are done this z5 it's our legitimacy
so finally already stands why it's equal to just the C ok e to the negative x to the fourth power
over 4 and we add one app to this and here is the general solution and we isolate the Y as well in
this case I like the wipers and I would like to look back to my initial value and plugging in the
sofa see here we know X is equal to 0 Y will be 3 so I would like to plug in 3 right here for this
Y and this is going to give us C I don't know yet times e to the negative x we know is 0 so we have
0 and then raised to the fourth power and over 4 at the end we add the 1 after that I'm going
to solve this real quick this is 3 it's equal to C is just e or everything all together is just
going to be 1 right so C times 1 and then we add a 1 after that so this is just 3 times C times
1/2 C plus 1 and then subtract 1 on both sides we know C it's going to give us 2 plugging 2 into the
C and we're done finally the answer is why it's equal to 2 and we have that e to the negative x to
the fourth power over 4 and then we add a regular one after this here is the answer and this is it