Okay, so for this question, this is a differential equations question and given 8 kg mass, right moves attached to a spring. So here's our mass, and that's attached to a spring that's hanging from the ceiling. And it's allowed to come to arrest and were given that the spring constant, um, or given that spring, constant is equal to 40. Right? So, um, is about 80 kg a spring constant k 0 to 40 and were also given, uh oh, 40 Newtons per meter. And we're given that the dampening constant B is equal to three Newtons, uh, Newton seconds per, uh per meter. And then, uh, time to equal zero. Um so those are all the factors or constant stuff was given and were also given an equation, which is the force to sign of two tee times co sign of two teeth. Basically, from all this, we have to find the amplitude and frequency of the steady state solution of, uh, this entire system here. Right. Okay, so the first thing we can do is we basically have to identify that this is basically a trigger metric identity, so we can actually simplify the function. F f t all the way down to sign of 40. Yeah. So once we do that, this actually becomes a whole lot simpler problem. And then we can set up our differential equation by saying 88 de tu y over dx squared. Uh, so second, even plus three dy DT it was supposed to be DT squared dy DT plus 40. Why? And these are our constants that we had appeared mhm. And, um, this is all going to be equal to a sign of 40 because that is also part of the system. So now we have to find the steady state solution. Uh, when we do that, we actually use the undetermined coefficients method, uh, by putting the solution in terms of or in the form of the sine cosine form. So when we do that, we have y P is equal to a co sign of 40 plus b sign of 40. I think that's gonna be the general form that we get this into. And then we have to plug that into this equation here. So we have to find, uh, second the first derivative of, uh, the steady state solution, which is just the YP part right once again, Uh, so when we find that we'll get, uh, during the coast on this negative sign so negative for a sign of 40 because that would be the inside of here using chain rule and endured of signs co sign so I can be co sign up for a key. And then your second derivative will similarly be basically the same things during Sinus Co sign Negative 16 a co sign 40 plus and then the during all of this. So because there's coastline is negative signs that this will change to a negative 16 b times sine of 40 mhm tonight, you basically take all of these and you plug them back into this equation here. And when you do that, you'll basically end up with eight times this. So it'll get eight times negative 16 a co sign of 40 minus 16, the sign of 40. And then you have plus three times yeah, divided teacher at the first derivative negative for a sign of 40 plus for be co sign of 40. And then you have plus 40 times Why? So I'll be plus 40 times a co sign 40 plus B uh, sorry. Yahiko Center 14 plus B sign of 14. Mhm. And then all that equal to sign a 14. So once we have this huge equation, uh, we can start simplifying it, uh, down, uh, right so we can go and start by multiplying everything inside. Uh, so if you put eight inside here, you get negative. 1 28 a co sign of 40 minus 1 28 b sign of 40 plus or it actually minus minus 12. A sign of 40 um, plus 12 b coastline of 14. Right when you multiply those two in and then plus 12 b co sign of 40 plus 40. A co sign of 14 plus 40 b sign of 40. All of that equal design of 14. So now all you have to do is take your co signs and take your signs. Right. So you have coastline here? Uh, cause I'm here cause I'm here and then you have sign here, sign here and sign here. And you basically have to combine all of those together into one term and do that, Um uh, by basically taking, uh, coastline of 40 out of the equation like that factoring it out and then having all of your inside Constance a negative 1. 28 a. Right, so that takes care of that term. And then here you have plus 12 B that takes care of that term. And then here you have plus 40 A. So when you add that you actually get in your negative 88 a And then you add to that your sign of fourty term, uh, which you will actually also take out. And then you have negative 1 28 B minus 12 A. And then you add into that plus 40 b, and this actually becomes negative 88 b And then you said all that equal to sign of 40. Now you can see that there is no co sign term on this side of the equation. So this has to be equal to zero. And so what that means is negative. 88 A. Since this can't be equal to zero negative 88 a plus 12 b has equal to zero and then coefficient for this sign in the 40 here is just equal to one. Then that basically also says negative 88 B minus 12 a this part as we equal to one. Right. So now all you have to do is solve for both of them, and, uh, easy way to do that is, um, first saw for B, um, on this side. Right. So you'll get 12 b equals 88 a and then B equals 88 a over 12. Um, so you do that you can, uh, simplify this a little bit, Uh, 2. 22 a over three. If you divide both sides by four and then you can plug that into this equation here and you'll get negative. 88 times 22 a over three, minus 12 A equals one. And then when you multiply that in, you'll get 88 times. 22. Uh, so you'll get negative. 19. 36/3, a minus 12, A. Equals one. Mhm. And then, um uh, 12. You can basically rewrite that as, um, uh, 36/3. All right. And so that way you can combine both of these together. So you have 36/3, and then you multiply 3/19, 36 a minus 36 a three, right, and then you'll just combine these two and then you'll get, uh, a equals Negative 3/19. 72 when you do this. So now that you have a all you have to do is plug that back into this equation here or any of them, and, uh, plug that back in to find B, and we know B is equal to 88 a over 12. So if we plug that back in, uh, B equals 22 a over three, then that must be is equal to 22 times like there's 3/19 72 all of us, all of the three. Um, when you do that, you'll get basically negative. 11. These stuff will cancel out over 986 So that will be your value for B. So now that you have both A and B, uh, you can plug those into your y p solution here. When you do that, you'll get y p equals negative. 3/19 72 co sign of fourty right minus 11/9, 86 times sine of 40. So now that we have that, that is a steady state solution. Now we have to find the amplitude, and you can find the amplitude of this, uh, amplitude of a form in a co sign X plus B co sign X is equal to, uh, square root of A or amplitude. My smaller case a a squared plus B squared. So that will be this square plus the square, so that when you do that, you'll get negative 3/19 72 squared plus negative 11/9 86 square, and I'll give you the square root of 4 93 over 1972. Don't just take my word for it. Go ahead and solve for yourself so you can see the answer. But that will be your amplitude, and that's how you find it. And then also, the question also asked us to find the frequency. So are the angular frequency. Um, so in both cases, frequency is basically going to warn over tea and angular frequency is equal to, um, the angler frequencies to pi over t. So for both cases, you have to find t and T is given to you by two pi, uh, over four. Because in this case, in this case, in our case, specifically, uh, the period is two pi over four. Um, because because it's being multiplied here on the inside by four. Uh, so full period would be to pie. So you divide that by four and you get to power for the same thing as part of the two. So that way you frequency would be won over pi over two or two over a pie. And that means also your angular frequency is equal to, uh to pi over pi over two, which is the same thing as for and both of these are in four times one over a 2nd 14 hertz, basically. So your frequency is equal to two over pie and angular frequency is equal to four times a second to negative form, and those are your answers for this system, and that's basically how you solve it.