00:01
This problem we need to use the force equation so what's the force equation? force is that mx dot dot plus k the spring constant x is equal to the force thing term okay so let's solve or let's put in the values so this is going to be 2x dot dot plus uh 32 x is equal to 80 to the 4t, cosine of 4t.
00:39
Okay.
00:42
And we can divide everything by 2.
00:44
So we get x.
00:46
Dot, dot, plus 16x is equal to 40.
00:53
E to the 4t, cosine of 40.
00:58
So the first thing we need to do is find a particular solution.
01:01
So in order to find a particular solution, let's look for something that can give you something of this form.
01:07
So let's look for let's choose a particular solution that so what can give you something of this form is if you have a constant times e to the 4t cosine of 4 plus and then usually it's nice to also have the other some extra term to cancel out stuff for t because cosine and sign are complementary so this might help you cancel out some things so let's look for something of this form because it's a and b now let's compute.
01:44
So x.
01:45
Dot is going to be 4a e to the 4t, cosine of 4t, minus 4a, e to the 4t, sine of 4t.
02:07
Okay, this is from the product rule of the first term, and then you get plus for b.
02:15
E to the 4 t sign of 4t and then you get plus 4b e to the 4t cosine of 4t okay and then we do it again well okay let's first simplify so this will be 4a plus 4b times e to the 4t cosine of 4t plus 4t plus 4b minus 4a e to the 4t sine of 4t.
03:05
Okay, now let's take two derivatives.
03:13
Note that this has the same form as this one just with the different constants.
03:19
So you can kind of guess what the pattern is going to be, but let's do this.
03:25
So this is going to be 16 times a plus b, e to the 4t, cosine of 4t, plus 16 b minus a.
03:46
So from this one, e to the 4t, cosine of 4t.
03:52
So you get cosine also from this terms, from this side of these two terms.
03:56
And then we have the so from this term we'll get a plus 16 b minus a e to the 4t sine of 4t which is this term and then we'll also get the other one where you do 16 b minus uh not sorry not b minus a uh b or a plus b.
04:34
4t cosine or not cosine of 4t so this is this one so again when you simplify these what you get you get that this is equal to 16 a plus b plus b minus a is going to be 2b e to the 4t cosine cosine of the other one we get plus 16 and then you get b minus a minus b minus b so it'll be minus 2a minus 2a e to the 4t sine of 4t okay so that's what we get in the end of this and now let's plug into this equation so we have x dot dot plus what is 16x 16x well what is this if you plug it everything in this is going to be this 32 b e to the 4t cosine of 4t minus 32a e to the 4t sine of 4t and then 16 times the other one so you get plus 16a, e to the 4t, cosine of 4t, plus 16x, 16b, e to the 4t, sign of 4t.
06:46
Okay, so this is, if we put all the like terms together, this is going to be 32b plus 16a, e to the 4t, e to the 4t, cosine of 4t plus 16b minus 32a, e to the 4t, sign of 40.
07:22
And we know that x dot dot plus 16x is equal to 40, e to the 4t, cosine of 4t.
07:32
So from this, we can match terms...