00:01
Okay, so to find the general solution of this problem, first let's find the corresponding homogeneous solution, or the corresponding complementary function.
00:10
So that's when like this is equal to zero.
00:17
So we first solve this problem here.
00:21
Now, it has the corresponding auxiliary equation, p of r, is equal to r minus 1, r plus 2.
00:29
We set that equal to 0.
00:31
So we have r is equal to 1 and negative 2.
00:35
Therefore, our complementary function, y, c of x, is going to be equal to.
00:42
Now we have c1, e to the x, plus c2e to the negative 2x.
00:49
Now we need to find the annihilator of the trial solution here.
00:55
So our annihilator, a of d, if we notice here, this has an r equal to 3, so our annihilator is going to take the form d minus r, or sorry, d minus r, which is just 3.
01:16
So we just have our annihilator is d minus 3.
01:19
Now we apply the annihilator to both sides of this equation here.
01:26
So if we do that, we have d minus or, yeah, d minus 3 times d minus 1 times d plus 2 of y is equal to 5e to the 3 or sorry.
01:45
And then a of d or d minus 3.
01:53
Let me move that a little over here.
01:57
D minus 3, 5 to the, oops, d minus 3 times 5e or of 5e to the 3x.
02:13
And then this, remember, it's called the annihilator because it annihilates that, and that just becomes 0.
02:20
So our general solution will be a solution of the form, d plus 2, y, is equal to 0.
02:28
So y is going to end up.
02:34
So first we have these two already...