 look at this damped motion. So if we look at the equation first that we have here, again it was x and t, so the damped motion meant that with my auxiliary equation I had the square root of a number less than zero. That was damped motion and our answer was in this format, remember it was x of t, it was e to the power negative gamma t plus c sub 1 cosine of gamma squared minus omega squared times t plus c sub 2 times the square root of gamma squared minus omega squared times t. So I'm not writing the x of t, I'm using the f of x, so all my t's have to change to x's and with my remember we set 2 times gamma equal to beta over m, so I'm just reintroducing this p now as the beta, I can't put beta into this program so I've just got p divided by 2m, that would be my negative gamma. If you just looked at what we have gamma times t and here we have omega squared which is then k over m minus omega squared minus gamma squared. And let's see what happens if we have a look at this system. So there we have as time goes our oscillatory motion gets damped over time. Now remember to get to this we introduce this new force into our simple harmonic oscillation k or x double prime plus omega squared x equals 0, we introduce this force that we called beta x prime and it was opposed to the direction of velocity, this force's direction was opposed to the direction of our velocity. And it's actually a very simplified way of seeing this, imagine force affriction of something being dragged across the floor, there's so many electromagnetic interactions and that was also a very simplified way of doing it, it's actually negative beta x prime and then there's another x squared term that you can bring into that, for instance when the velocity is very small we look at that squared term and that's when you drop a ball being say into something viscous like honey, the velocities they are a lot smaller than we don't look at the beta x prime component, we look at the other component. Be that as it may, this is a mathematics lecture so we just introduce that single term for the force of affriction I should say and that was beta x prime. And that was though just a measure of, a measure of the defriction, so p here or my beta, if I let the friction get larger and larger you see how very quickly now suddenly we have the damping, if I make the friction less and less and less and less you see the damping gets less, that's what we would expect, if I make the mass larger I use a larger mass, that means my period this is going to get longer, see how the amplitude doesn't change, the frequency goes down, the period gets longer. What about my C sub 1 and C sub 2, see what that does, we're playing around with amplitude here. Now it is moving a bit with C sub 1 and C sub 2, you can see the amplitude change but you can also see a shift in the on the x axis, does that mean that the period is changing, does that mean that the frequency is changing, well let's have a look, yeah I have period, let's get rid of our function there, we have period and we have frequency. So let's first look at what happens to the period, if I now I'm hanging a mass still on a spring and I'm living it oscillate up and down or I have a mass hanging from a string, it's a pendulum so it's swinging back and forth, now if I change the mass indeed as the mass gets larger I am indeed making the period longer and if we zoom in on the frequency, yes indeed as they make the mass larger the frequency comes down, now of course the opposite happening for the spring constant here but remember we said if we let C sub 1 and C sub 2 change certainly the amplitude change but the frequency and the period change well let's play with it, nothing happens whatsoever, nothing happens whatsoever so those constants just set my initial conditions, so I can give it a push in a certain direction, I can give it some velocity but it's not going to make a difference again, those are just initial conditions, they do not make a difference to the period or to the, let's zoom our period or until we get the frequency again, look as I play with it, the period and the frequency does not change, period and frequency is going to change if I put a different mass on it, ok so amplitude, let's get back to the amplitude let's try and add that, remember amplitude, let's call the amplitude, let's call that the S of X for adding another variable there and that was the square root of A squared plus B squared, B squared there we go, so let's get rid of these, so now we are just looking at amplitude, so as I play around with the, as I play around with these sliders you can see that A and B changing, my amplitude is going to change but certainly the period, there we go with the period again though, it does not change by this initial conditions so I can pull out once again, I can pull out that mass at a bigger angle as it hangs from the string, I am not and I can give it any kind of initial condition, I am going to change the amplitude for sure but I am not changing the frequency and I am definitely not changing therefore the frequency, excellent you can use these equations and we can predict what is going to happen, do that experiment and we will see within the margins of error and with the fact that we are fudging a bit this global force of friction into one simple term opposing the velocity, the direction of the velocity but it gives us a very good intuition as to what is going to happen, a very good approximation of what is going to happen, so learning these differential equations certainly helps us a lot in physics.