00:01
Hello and welcome to problem 4 of chapter 3 section 7.
00:03
Here we're asked to use the method of variation of parameters to find a particular solution to the given differential equation.
00:09
Our differential equation in this case is 4 y double prime minus 4 y prime plus y equals 16 e to the t over 2.
00:16
So we start off by finding solutions to the characteristic equation and then finding this capital y of t turn, which is what variation of parameters is.
00:27
So i'll start off with, do it red, 4r squared minus 4r plus 1, 0.
00:36
It's factor 2r minus 1 squared, 0.
00:41
So, yeah, so we'll have double root of, let's see, r equals 1⁄2, i believe.
00:51
Yes.
00:52
So that means our two solutions will be e to the t over 2.
01:00
T e to the t over 2 okay so we can plug this into our equation for y of t but first we should find the ron skin of these because that is necessary for this y of t term so the run skin is just y1 times y 2 prime minus y 2 times y 1 times y 1 prime this is just from the 2 by 2 determinant of y next derivatives so we can just plug in terms.
01:33
We'll have e to the t over two times, well, the derivative of dt of t, e to the t over two.
01:44
We'll do that in a sec, but minus t e to the t over two over two times e to the t over two.
01:54
Okay, so we're almost there, but we'll have to simplify.
01:57
And we first have to find the derivative of t e to the t over two by a product rule.
02:03
So let's just do it here real quick.
02:08
So it's t times the derivative of e to t over two.
02:13
So it's e to the t over two over two.
02:18
Then we'll just add this to e to the t over two.
02:22
Okay.
02:23
And when this thing is all multiplied by e to the t over two as in there, we'll just get e to the t.
02:30
So we can simplify that there.
02:33
All right.
02:34
And then we will be subtracting this from another term, which is right here.
02:43
And if we can, upon inspection, we see that that will be simplified to t .e.
02:49
T over 2, which is exactly what we have here.
02:53
So this will cancel and will just be left with e to the t.
03:04
Equals e to the t...