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8+29 + 5q = 03€ 20 Jt constant coefficientlinear differentialequationand 9(02 = 0 ve 9' (0) = 10 initialvalues are given Defined forthis circuit Eindthe ...

Question

8+29 + 5q = 03€ 20 Jt constant coefficientlinear differentialequationand 9(02 = 0 ve 9' (0) = 10 initialvalues are given Defined forthis circuit Eindthe Lapbae transfom

8+29 + 5q = 03€ 20 Jt constant coefficientlinear differentialequationand 9(02 = 0 ve 9' (0) = 10 initialvalues are given Defined forthis circuit Eindthe Lapbae transfom



Answers

The current $I(t)$ in an $R L C$ series circuit is governed by the initial value problem $$I^{\prime \prime}(t)+2 I^{\prime}(t)+2 I(t)=g(t)$$
$$I(0)=10, \quad I^{\prime}(0)=0$$
where
$$g(t) :=\left\{\begin{array}{ll}{20,} & {0< t <3 \pi} \\ {0,} & {3 \pi< t <4 \pi} \\ {20,} & {4 \pi< t}\end{array}\right.$$

And today will be solving a problem which states that considered RC circuit with our sequitur. Two homes sees tickle toe one a fraud and 80 is equality 10 times. Cose entreaty boats Assuming that that there's a charge of one capacitance initially determine the current in the circuit for t greater dinner equal to zero. So the first thing we'll do is figure we're gonna write down our defensive equation for R C. That is cute. Prime plus one over RC que that is over is equal to e t over our you know, our secret, too. Season 1/8 e t. C quarter Sen Times cool saying ST T and we know that there's gonna be a charge off one capacitance So cube zero secret one. So now we don't know everything this plug in plug it in to our defensively question. So we have this cute prime plus one over two times 18 terms que their secret is sand co sign three t all over, too. So up to simple playing, simplifying everything what we'll get q prime plus for a que the secret of five co sign 32 So what we do now is that I'm gonna write e two. Integral of what is our coefficient of Q? Which is for And we do that respect to DT that would equal to eat to the 40. So we do now. It's going to play. Needed a fourty toward differentially equation. So what will have each of the 40 Temps cute prime plus for each of 40? It saps que Dacic would've five times need to the 40 Taps Coast and a three tier so basically the left side that is just the anti derivative for you to the 40 times. Q. So to get Cuba itself, I would need to do is take dinner. Go both sides. So what we have in a love site, it's easy to 40 times. Q. Right said we have Dana, Girl of five Times Eater 40 Tom's co sign Treaty de to So using integration by parts. What will have with the to 40 times Q. Does he quit a five Sam's Do You Over 25 eat a 42 signed treaty plus four over 25 each of 40 times co sign treaty that being added by sea. So we simplified that right side. We'll get Street, fifth each to 40. Scientific Teoh plus 4/5 each and 40. That's Cho Sang Treaty plus C. So we divide both sides by each of the 40. What would get Street fifth? So I'm sad. Treaty Plus 4/5 Co sign Treaty plus C times. It's a negative for two. So we're not done yet because we have see, as an unknown we do know is that Q zero secret one. So we could do is we'll plug it one in four Q is there and for tear, so it's one equal. The Fifth Sign ST Time zero plus 4/5 co sign Be Time zero. Let's See, I Was You tonight of four times zero. So after simplifying everything and solving for see what we'll get, this seasick would have won over five. So cue secret. 3/5 Sane 32 plus four or five Co sign Treaty Plus went fifth, the to the negative 40. Now do you ask us to find a current or I have tea that is just equal to the derivative off Q of T. So that it's just the derivative for all of this sequel. 9/5 Coast entreaty minus so over five. Sign treaty minus for over five a times B to the negative 40. So the current for DiSarcina circuit isn't 9/5 co sentry T minus 12/5 sign three T minus 4/5 times E to note off for tea.

And today will be solving a problem which states that considered RC circuit with our sequitur. Two homes sees tickle toe one a fraud and 80 is equality 10 times. Cose entreaty boats Assuming that that there's a charge of one capacitance initially determine the current in the circuit for t greater dinner equal to zero. So the first thing we'll do is figure we're gonna write down our defensive equation for R C. That is cute. Prime plus one over RC que that is over is equal to e t over our you know, our secret, too. Season 1/8 e t. C quarter Sen Times cool saying ST T and we know that there's gonna be a charge off one capacitance So cube zero secret one. So now we don't know everything this plug in plug it in to our defensively question. So we have this cute prime plus one over two times 18 terms que their secret is sand co sign three t all over, too. So up to simple playing, simplifying everything what we'll get q prime plus for a que the secret of five co sign 32 So what we do now is that I'm gonna write e two. Integral of what is our coefficient of Q? Which is for And we do that respect to DT that would equal to eat to the 40. So we do now. It's going to play. Needed a fourty toward differentially equation. So what will have each of the 40 Temps cute prime plus for each of 40? It saps que Dacic would've five times need to the 40 Taps Coast and a three tier so basically the left side that is just the anti derivative for you to the 40 times. Q. So to get Cuba itself, I would need to do is take dinner. Go both sides. So what we have in a love site, it's easy to 40 times. Q. Right said we have Dana, Girl of five Times Eater 40 Tom's co sign Treaty de to So using integration by parts. What will have with the to 40 times Q. Does he quit a five Sam's Do You Over 25 eat a 42 signed treaty plus four over 25 each of 40 times co sign treaty that being added by sea. So we simplified that right side. We'll get Street, fifth each to 40. Scientific Teoh plus 4/5 each and 40. That's Cho Sang Treaty plus C. So we divide both sides by each of the 40. What would get Street fifth? So I'm sad. Treaty Plus 4/5 Co sign Treaty plus C times. It's a negative for two. So we're not done yet because we have see, as an unknown we do know is that Q zero secret one. So we could do is we'll plug it one in four Q is there and for tear, so it's one equal. The Fifth Sign ST Time zero plus 4/5 co sign Be Time zero. Let's See, I Was You tonight of four times zero. So after simplifying everything and solving for see what we'll get, this seasick would have won over five. So cue secret. 3/5 Sane 32 plus four or five Co sign Treaty Plus went fifth, the to the negative 40. Now do you ask us to find a current or I have tea that is just equal to the derivative off Q of T. So that it's just the derivative for all of this sequel. 9/5 Coast entreaty minus so over five. Sign treaty minus for over five a times B to the negative 40. So the current for DiSarcina circuit isn't 9/5 co sentry T minus 12/5 sign three T minus 4/5 times E to note off for tea.

Okay, let's first make a little changes to this system of linear equations. Okay, so the next step, we can usually turn it into a metrics. For the first equation, I want to move the ice rate to the left side. So I have I I one plus I to minus I. Three plus I. Four. He goes to zero for the For the 2nd equation. Okay, Let's move 40 to the right side. So negative 200. I won Last 80. I too equals to 40 For the 3rd equation. So three, plus 40 is 400. And we move 400 to the right side. So we have Native 80. I too minus 20 I. Three yes minus 400. Okay, for the last one. so we have 200 -200. I won plus 70 by four because to It was 280. Yeah. Okay, and actually we can further simplify this. The third, the second one. The third one and the fourth one. Okay, for the second equation for this equation, what we can do is we can divide it ah 40. Okay, so this is the biggest common factors of the left side and right side. Okay, we're divided by 40 uh by the both sides of the equation. So the left side we have nah negative 95. Here is to sorry, here is to and the right side just become just becomes one. Okay, okay, and uh For the third equation we can divide it by 22. The both sides of the equation Are negative 20, that's divided by negative 20 to turn all the equation to positive. So this So the first will become four and the second coefficient is positive. Okay, just positive one. So I just wrote positive plus. Okay, from the right side. Okay, after the division, What left is 20? So here I wrote 20. Okay, for the For the last one, what we can do is we divided by 10 to the both sides of the equation. So it's very easy. It would just erase 10 at every coefficient and also at the right side. Now the equation systems looks much much more easier than it is before. So now let's turn it into ah into a metrics. So that's the here I have. Okay, so the first column is I want, so I have 1 -5 and negative 20 15 95 0 and 1920 1 -50 and -20. And ah for the second column is I too, so I have one two for it is zero 1240 for the third column I have I threes -101 zero. For the 4th column I have one 00 and seven. And then the right side is 0120 and eight. Okay so mm. Okay so and after the solving process, what we get for these metrics is Next one. It goes to 90 -51 divided by 2-9. Okay. The second one is X two is -3 divided by two July. and the X three is sorry to to night 52 divided by 2- nine. And the X four. Okay maybe uh ruled it down here. X four X four is 100 sticks team Divided by 2-9. Okay so this uh solving process is very uh difficult and tedious. So if you want to find find how to do it, maybe you can try to use your graphic calculator.

So here's the question is I equals to a note into the power minus divided balutis. The given value that I notice it closer to a in case a let's part let's solve the first part. In a to calculate I. So I will because I know I notice given us to MPR multiplied with the to the power of minus E. So time is given us. I am is given a zero my 0.3. So it will be 0.3 divided by R. C. For assistance is given a six multiplied with 10 to depart of minus five. Home military with capacitance. Okay persistence is given as the persistence is given as five multiplied with and to the power of minus seven. Parade. So The resistance is not here it is positive X. multiplied with and to the power of five and capacitance is After solving we will get eyes equals two would be very man mm And this is the required value of the first part. Very very when part B. We need to calculate the I divided by D. T. D. I did whatever deity this will because two minus. I know you were there were R. C. It will be part of many still developed. So when these equals to 0.3 seconds Then d. 80 where is my duty Active equals to 0.3 seconds. This will be close to minus motivated man minus two. Divided by R. C. Z equals 01 thing when compared with the to the power of dynasty. It is the open three. Very very RC. Which is close to zero Upon solving we will get -20 divided by the M. P. R. Person. This is the solution of the 2nd part. Let's move to pardon artsy here and it took calculate I. And easy goes to Your appointee 1 2nd We have i.e. z equals two. two multiplied with the to the power of minus You're open. Tree one divided by zero country. The dispute give us a value 5.8 divided by him. The mhm mps. This is the pollution of the party.


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