00:01
We were asked to solve this differential equation, and note that the general solution will be the complementary solution plus the particular solution, where for the complementary solution, we're just solving the homogeneous differential equation.
00:24
So just this differential equation.
00:27
And of course, the auxiliary equation is y plus, sorry, r squared plus 4 equals 0.
00:35
So you get that r equals plus or minus 2i.
00:41
And then this means that y sub c, this is c1 times cosine 2x plus c2 times sine 2x.
01:01
Ok.
01:02
And then now for y sub p, right, what should our guess be? so y sub p is based off of this function, but then it's also based off of what the complementary solution is as well.
01:17
So that's why you always want to find y sub c first, because that influences what y sub p will be.
01:24
So anyway, normally if we just saw, right, this function, then we would guess a times cosine of 2x plus b times sine of 2x, and then plus capital c times e to the x.
01:50
And the reason why we include sine and cosine is because when we take the derivative of sine, we see that we get cosine.
01:58
So it's just it's important to include all of the linearly independent functions that will pop out once you take the derivative.
02:07
However, this is not the correct guess for y sub p.
02:11
And the reason is because this function right here is already within the complementary solution.
02:19
So and you never want there to be any overlap between the two.
02:24
So that's why we multiply by x.
02:27
So if you hadn't multiplied by x, then, you know, what we originally had, it would have just solved this homogeneous equation.
02:39
Anyway, now we have to find out what a, b, and c are.
02:43
And in order to do that, we'll take y sub p, and then we'll plug it into the differential equation.
02:49
But in order to do that, we have to calculate what the second derivative is.
02:54
And in order to do that, we have to calculate what the first derivative is.
03:07
Okay, so we've calculated the second derivative...