To solve this differential equation, you need to figure out which functions are their own first derivatives with respect to time

Thats certainly true of this function, but I didnt ask for one solution, I asked for the entire family of solutions

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

This one here is not quite right

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

The number e actually has a special name because of exactly the property we were playing with in this quiz problem

I said e^t is a function that is its own derivative with respect to time

Thats why e has that special name

That is, the slope of the function e^t is equal to the height of the function e^t at all points

So its derivative is equal to the function itself

The solution to the second problem involves going back to the drawing that we did in the unit

We have a mass on a spring, and the force balance looks like this

And if those two forces are out of balance, then the mass will accelerate: F = ma

Now the only thing thats hard to get right here is really the minus sign

And if you look at the drawing, you see that x is measured pointing down, and mg is pointing down

So mg will make x bigger, whereas kx, the force pointing up, will make x smaller

And thats how I remember how to get that sign right

Then the last piece is that a is x

And if you put all that together, you can see that this one is the right answer here