let's solve this separable differential equation we have dy/dx is equal to 8x plus third power times e
to the negative 2y power and we also have Y of one is e to the 0 so be sure to stop at the constant
all right let's go ahead multiply the X on both sides and you see here with the y part right so
let's go ahead and divide both sides by e to the negative two Y on the left hand side I'll put the
stands 1 over e to e then I could keep - why right before integrate this let me write this down as e
to the past get 2y and then this is it's a dy and this is equal to 8 X to the 3rd power and
we have 2 DX and now we can go ahead integrate integrate is it ok integral e to the something
this is just a constant times y you notice it's going to be just e to the same thing - Y button
at the relative to Y is 2 divided by 2 right because we're doing antiderivative 1/2 e to
the 2y if you differentiate it you'll get that back it's great don't worry about a + C let me
just read this x equals 2 and here we have 8 X to the 3rd power you add 1 to the power
which is X to the fourth power right so let me just show you guys the steps plus 1 which is
going to give us 4 and D PI D PI the new power 8 divided by 4 is 2 so 2 X to the fourth power and
right here I'm just going to put down plus c1 maybe c1 c2 to us already matter up to you and
in this case I can just plug in the initial condition right here and solve for the see
if you were like and I think I'll just do that you can also soft white first and that and yet
so fancy in this case doesn't really matter so let's do that right here we know when X is equal
to 1 Y will be 0 so in here we will have 1/2 e is just e and then 2 and the y is 0 and this is
equal to 2 times X is 1 so be sure when you do this kind of things plugging the right number
with the right variable plus I just missed e huh I don't know c1 so let me just write RC right right
here this is 1/2 and into the zeros power I pretty much 2 times 0 is just 1 so it's 1/2 and 1 to the
first power is 1 times 2 is just 2 and we see to that and of course we -2 on both sides okay so C
will be 1/2 minus 4 over 2 is negative 3 over 2 let's see so here you know you can't just write
this down as 1/2 inches ready stance 1/2 each will be 2 why it's equal to 2x to the 4th power
and the C is that negative 3 over 2 and now as I said I would like to isolate the Y to do so let's
multiply everything by 2 so that this and now cancel and we will have e to the 2 y and this
is equal to distribute distribute 2 times 2 is 4 X to the 4th power 2 times - just minus 3 and
at the end I need to carry off the e the whites and the powers let's go ahead and take the Ln on
both sides and I'm not integrating this is just a regular arrow on both sides situation parentheses
no absolute value okay anyway they cancel and we will have 2y equals to Ln parentheses 4x to the
fourth power minus 3 finally multiply by post by 1/2 and we're done so already stun nicely for you
guys why it's equal to 1/2 Ln parentheses for extra force power minus 3 this is it