2nd order linear homogeneous differential equations 1

we'll now move from the world of first-order differential equations to the world of second-order differential equations so what does that mean that means that we're it's now going to start involving the second derivative and the first class that I'm going to show you and this is probably the most useful class when you're studying classical physics are linear second order differential equations so what is a linear second order differential equation so I think I touched on it a little bit in the in our very first intro video but it's something that looks like this if I have a of X so some function only of x times the second derivative of Y with respect to X plus B of x times the first derivative of Y with respect to X plus C of X times y is equal to some function just of that's only a function of X so just to review our terminology why is the second order because the highest derivative here is the second derivative so that makes it second order and what makes it linear well all of the coefficients on and I want to be careful with the term coefficients because traditionally we view coefficients as always being constants but here we have con we have we have functions of X as coefficients so in order for this to be a linear differential equation a of X B of X C of X and D of X they all have to be functions only of X as I've drawn it here and now before I we start trying to solve this generally we'll we'll do a special case of this where ABC are constants and D is 0 so what will that what look like so I can just rewrite that is a so that now a is not a function anymore it's just a number a times the second derivative of Y with respect to X plus B times the first derivative plus C times y and instead of having just a fourth constant as instead of d of X I'm just going to set that equal to Z and by setting this equal to zero I have now introduced you to the other form of homogeneous differential equation and this one is called homogeneous and it's and I haven't made the connection yet on how these second order differential equations are related to the first-order ones that I just introduced these other homogeneous differential equations I introduced you to I think they just happen to have the same name even though they're not that related so the reason why this one is called homogeneous is because you have it equal to zero so this is what makes it homogeneous and actually I do see more of a connection between this type of equation and milk where all the fat is spread out because if you think about it no matter the solution for all homogeneous equations or when you kind of apply when you kind of solve the equation they always equal zero so they're they're homogenized I guess it's the best way that I I can draw any kind of parallel so we could call this a second order right second order linear right because a B and C definitely are functions just of well they're not even functions of X or Y they're just constants so second order linear homogeneous because they equals zero differential equations and I think you'll see that these are in some ways are the most fun differential equations to solve and actually often the most useful because a lot of the applications in classical mechanics this is all you need to solve but they're the most fun to solve because they all boil down to algebra two problems and I'll touch on that in a second but let's just think about this a little bit think about what the properties of these solutions might be let me just throw out something let's say that I don't know let's say that G let's say that G of X is a solution G of X is a solution so that means that a times G prime prime plus B times G prime plus C times G is equal to zero right that these mean the same thing now my question to you is what if I have some constant times G is that still a solution so let's say so my question is is let's say some constant C one G X C one times G is this a solution well let's try it out let's substitute this into our original equation so a times the second derivative of this would just be and I'll switch colors here I'll switch maybe let me switch to brown so a times the second derivative of this would be the constant every time you take a derivative the constant just carries over right so that'll just be a times C one G prime prime plus the same thing for the first derivative B times C one G prime plus C times and this C is different than the C one C times G and let's see whether this is equal to zero so we could factor out that C one constant and we get C one times a G prime prime plus B G prime plus C G and lo and behold we already know because we know that G of X is a solution we know that this is true so this is going to be equal to zero right because G is a solution so if this is 0 C 1 times 0 is going to be equal to 0 so this expression up here is also equal to 0 or another way to view it is that if G is a solution to this second-order linear homogeneous differential equation then some constant times G is also a solution so this is also a solution to the differential equation and then the next property I want to show you and this is all going someplace don't worry the next question I want to ask you if okay we know that G of X is a solution to the differential equation what if I would also tell you that H of X H of X is also a solution also a solution so my question to you is is G of X + H of X a solution if you add these two two functions that are both solutions if you add them together is that still a solution of our original differential equation well let's substitute this whole thing into our original differential equation right so we'll have a times the second derivative of this thing well that's straighten straightforward enough that's just G prime prime plus h prime prime plus B times the first derivative of this thing G prime plus h prime plus C times this function G Plus H and now what can we do let's distribute all of these constants we get a times G prime prime plus a times H prime prime plus B times the first derivative of G plus B times the first derivative of H plus C times G plus C times H and now we can rearrange them we get a let's take this one let's take all the G terms a times the second derivative of G plus B times the first derivative plus C times G that's these three terms plus a times the second derivative of H plus B times first derivative plus C times H and now we know that both G and H are solutions of the original differential equation so by definition well yeah by definition if this is if G is a solution of the original differential equation and this was the left-hand side of that differential equation this is going to be equal to zero this is going to be equal to zero and so is this going to be equal to zero so we've shown that this whole expression is equal to zero so if you if G is a solution of the differential equation of this second-order linear homogeneous differential equation and H is also a solution then if you were to add them together there ought that that the sum of them is also solution so in general if we show that G is a solution and H is a solution you can add them and we showed before that any constant times them is also solution so you could also say that some constant times G of X plus some constant times H of X is also a solution and you know maybe that the constant in one of the cases is 0 or something I don't know but anyway these are useful properties for - maybe internalize for second order homogeneous linear differential equations and in the next video we're actually going to apply these properties to figure out the solutions for these and you'll see that they're they're actually straightforward I would say a lot easier than what we did when the the previous first-order homogeneous differential equations or the exact equations this is much much easier I'll see you in the next video