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Write the differential equation corresponding to periodicmotions by defining a system that makes a SIMPLE HARMONIC MOTION?and its solution;In terms of complex numbe...

Question

Write the differential equation corresponding to periodicmotions by defining a system that makes a SIMPLE HARMONIC MOTION?and its solution;In terms of complex numbers) In terms of trigometric functions) In terms of phase angle express

Write the differential equation corresponding to periodic motions by defining a system that makes a SIMPLE HARMONIC MOTION? and its solution; In terms of complex numbers ) In terms of trigometric functions ) In terms of phase angle express



Answers

Find the particular solution to the differential equation $\frac{d u}{d t}=\tan u$ that passes through $\left(1, \frac{\pi}{2}\right), \quad$ given that $u=\sin ^{-1}\left(e^{C+t}\right)$ is a general solution.

So in this question we have to give an example of a differential equation. Who is? Uh we're just having uh techno metric function as a solution. Like so let's take an example of a differential equation. Day, bye bye. Dx is equal to cynics. So the differential equation is also techno metric here, Right? We'll try to find the solution of the equation and we'll verify this so we can write it as diva is equal to sine X. Dx and we can integrate both sides. So when we integrate we get y equals two minus kazaks plus C. Like this is the general solution for this for this given differential equation. Right? So we can see that this is the general solution. So by changing the value of C we will get different different solutions. Total differential equation. Right? So here we can see that the general solution is a techno metric solution. Like so they've given this example is very famous.

Well, what we have guilty it's a function of time is equal to how can pee times U to the power minus r T divided by twice off l Times. Go sign of Omega T and then we have l de to kill. Divided by DT Square place are Do you can do by the by DT Bless Q. Divided by C, is equal to zero and also decoder divided by DT, which is a differential off kill equals Q P in two minus are divided by two l into E to the power minus are TV body by two. AL are tee divided My to L. Times Co Sign of Omega T Times. Call sign off Omega T Less Omega signed Omega T right and then different shooting again we have. Let's different your piece again Then we have D to the square. Cumin divided by DT Square is equal to minus. Q P. Into minus already mighty by two l minus are divided by two l into e to the power minus rt. Divided by to hell into are divided by two l times. Go sign off Omega T Ah Bliss Omega Sign. We'll make a teeth. We'll make, uh, t right. And then we have the older terram less e to the power minus are tee and divided by to al into minus are omega divided mine to help sine omega t less omega Square co sign of Omega T. Okay. And then our next step is well, after, um, some simplification we get de square he owned divided by DT square is equal to ah, minus cume Who times e to the power minus rt. Divided by two l rt divided by two l into, um minus are square divided by four elsewhere. Uh, go sign off. Omega T minus to our omega Divided by two l Time. Sign off. Omega T Bless Omega Square. No sign of Omega T. Right. Okay. No. From equation number one. Let's say this is our equity number one. Uh, this one is He could remember Want no. From this equation, we have let me write down here. So from a question number one from equation one, we have minus pure p times you to the paller minus rt. Divided by two l into are scarier. Divided by four l square. Are times go sign off. Omega T minus two are omega divided by, uh, myself Adul into sine omega t sine omega T Let's omega Square uncle sign of Omega T all rights. It's let me write this are square. I know these two drums are not here. Let me right. Don't again. Excuse me. So mm. We have our square divided by four else critical sine omega t uh, bless. Uh, plus the next Rahm is, um l Omega Square designer for make a T minus. Go sign off. Omega T, divided by C, is equal to zero. So this is from the creation number one right and minus can be times e to the power minus rt. Divided by two. L, uh, cannot be equal to zero and then for will sign off omega t co sign off Omega T Into are square divided by for L Bliss L Omega Square minus one. Divided by C is equal to zero. I'd call sign Omega T It's common here. No, we'll make a square. Uh, here. Go sign Omega T, um is not equal to zero. I did it for we have our screener divided by for l less 10 times Omega screen minus one, divided by C. equals zero. And now Omega Square is equal to four. L minus are square, see divided by for elsewhere. See and ends Omega is equal to screen Rudolph for l minus R squared C divided by two l, uh, to now to every have four elsewhere. See? So then two, then l and then we have rootsy

May have. Do you have a low upon the Squire? That equals integration sign. T. Disquiet into T dark dictator. Hence it is integration W. To the par minus two dW. That equals duration data into science. Peter esquire the theater. This is -1 upon W. Which is equal to have integration. Find your dessert. Because we have assumed the task whereas zero or two Theta D. Theta becomes dessert. Now this become -1 upon w. Which is equal to half and do minus cost jade. Last see That equals -1 upon W. Which is equal to have minus cost to die squad plastic. See no from that we have W. Of Tita Equal -1. A bomb. 0.5 into minus cost. Do you guys quiet plus C. Or W zero? Equal minus one upon 0.5 minus cause zero plus C. So further these equals one is equal to -1 upon 0.5 Into -1 Plus C. R c equal minus one plus 0.5. That equal to -0.5. So W of tita becomes minus one upon 0.5 minus cost tita squired -0.5. Or further is the richness W of tha tha that is equal to once upon 0.5 Cost Theatre Square last 0.5. That is equal to then a barn five cost three to Squire plus five. That is equal to twice upon cause theta squared plus one. Hence We have w of theater that equals two twice upon Cause Tita is quite a plus one. This is our answer.

Today we're going bizarre robber number 47 from the section chapter revealed here The given different medication is why don't we less plus why it calls one by say next. So first we will find the gender and in particular, pollution for general solution can be replaced by double suture The square be square equals zero d squared equals minus four because plus or minus I Soviet good complementary function as see, even cause eggs plus said do fine they By using method off variation of parameters, we can find particular solution that is like be off X can be Yeah Why are bless be like so fast you'll find doubly which is because while you were by the by awareness call six by the is sign If my one dices my nothing nix by progressive cause X which is getting us far so you and reminisce in bigger minus like through fr affects by W integral minus I Next f of X iss one by side Next inter one but you're forgetting us Integral minus one, because minus X similarly b equals in bigger why were have of X during by W, which you will be begging us in Decorah by Brunnis If I Article six Indo one by sign except for fix by number You This is Ricardo in bigger car text, which is because love more fine x So the total solution is vice. The press might be why c plus like be busy is because see you are course Plessy. Booth. Dynex, My enough x y one Class B by do E by. Do you? That is the solution that send off a question. Thank you.


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