00:01
Okay, question 9, we're going to be solving a non -homogeneous differential equation.
00:04
So to do this, we're first going to find the general solution to the homogeneous differential equation.
00:11
So that's just going to be the same thing, but the external function is going to be equal to zero.
00:15
And then we're going to add that solution to our particular solution.
00:20
So first, let's work on getting this solution to the homogeneous differential equation.
00:24
We're going to be guessing e to the lambda t.
00:28
And when we do so, we're going to get lambda squared minus 4.
00:32
Equals 0.
00:34
And this has a simple solution, which is when we factor, it's going to be lambda plus 2, lambda minus 2 equals 0.
00:41
And this is going to give us our solutions for lambda, two solutions actually, lambda 1, lambda 2.
00:47
And these two solutions are negative 2 and positive 2.
00:53
And from this, since these two are different, and wait first, i'm going to actually write the solutions for homogeneous and the solutions are respectively, e to the negative 2t and e to the 2t.
01:04
And since these two are linearly independent, we can make our general solution for our homogeneous differential equation.
01:10
So homogeneous general solution is equal to c1 times e to the negative 2t plus c2 times e to the negative or positive 2t.
01:24
Okay.
01:26
Now that we have our homogeneous general solution, let's go ahead and work on our particular solution.
01:30
So i'll do that on the right side.
01:33
And i'm going to rewrite the differential equation as a linear operator being applied to a function, giving us the output to d cubed.
01:41
So it's going to be something like this.
01:43
Linear operator applied to the function, giving us 2d2.
01:47
And what the linear operator is actually doing is applying the second derivative of the function.
01:53
So d squared over d cubed, or d squared, d t squared, minus four times the original function.
02:02
Okay, so when we input this into, this is going to be our guess, we have to figure.
02:07
Out what our guest should be and our guest should give us 2 t cubed so going over to my sheet over here we're going to be working our external function is a polynomial of degree n and the degree is 3 and our our guess should look something like this and instead of using a sub n i'm just going to be using different variables so like instead of a sub 1 a sub 2 i'm going to be using ab cd and when we do that for this uh polynomial we're going to get a t cubed starting at cube because that's our highest degree, plus, oops, a t, q, plus bt squared, plus c, plus ct, plus d.
02:55
Okay.
02:57
Now, let's apply the linear operator to our guess.
03:01
So we're going to be doing the second derivative of this function, and that's going to be, so the second derivative of this was, is going to be 6a -t, 6a -t, the second derivative of this is just 2.
03:14
B, this is going to be zero and this is also going to be zero.
03:18
So we'd have to include that right now.
03:20
And we're going to be subtracting four times the original input.
03:23
So minus 4...