1
00:00:03,000 --> 00:00:11,000
To solve this differential equation, you need to figure out which functions are their own first derivatives with respect to time
2
00:00:11,000 --> 00:00:18,000
Thats certainly true of this function, but I didnt ask for one solution, I asked for the entire family of solutions
3
00:00:18,000 --> 00:00:27,000
The correct answer there is this one, and you can check that by plugging in: the derivative of k_1*e^t is k_1*e^t
4
00:00:27,000 --> 00:00:30,000
This one here is not quite right
5
00:00:30,000 --> 00:00:38,000
If you take its first derivative with respect to time, you get k_1*e^t, which is not the same as k_1*e^t + k_2
6
00:00:38,000 --> 00:00:44,000
The number e actually has a special name because of exactly the property we were playing with in this quiz problem
7
00:00:44,000 --> 00:00:50,000
I said e^t is a function that is its own derivative with respect to time
8
00:00:50,000 --> 00:00:52,000
Thats why e has that special name
9
00:00:52,000 --> 00:01:00,000
That is, the slope of the function e^t is equal to the height of the function e^t at all points
10
00:01:00,000 --> 00:01:04,000
So its derivative is equal to the function itself
11
00:01:04,000 --> 00:01:10,000
The solution to the second problem involves going back to the drawing that we did in the unit
12
00:01:10,000 --> 00:01:16,000
We have a mass on a spring, and the force balance looks like this
13
00:01:16,000 --> 00:01:21,000
And if those two forces are out of balance, then the mass will accelerate: F = ma
14
00:01:21,000 --> 00:01:24,000
Now the only thing thats hard to get right here is really the minus sign
15
00:01:24,000 --> 00:01:31,000
And if you look at the drawing, you see that x is measured pointing down, and mg is pointing down
16
00:01:31,000 --> 00:01:36,000
So mg will make x bigger, whereas kx, the force pointing up, will make x smaller
17
00:01:36,000 --> 00:01:39,000
And thats how I remember how to get that sign right
18
00:01:39,000 --> 00:01:42,000
Then the last piece is that a is x
19
00:01:42,000 --> 00:01:47,000
And if you put all that together, you can see that this one is the right answer here