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In the circuit shown in Figure 4, a generator supplies a voltage of $ E(t) = 40 \sin 60t $ volts, the inductance is 1 H, the resistance is $ 20 \Omega, $ and $ I(0) = 1 A. $

(a) Find $ I(t). $

(b) Use a graphing device to draw the graph of the current function.

a) $$

I(t)=\frac{1}{5}\left(\sin (60 t)-3 \cos (60 t)+8 e^{-20 t}\right)

$$

b) $$

I(t=0.1 s)=-0.41 \mathrm{A}

$$

c) $$

I(t)=\frac{\sin (60 t)-3 \cos (60 t)+8 e^{-20 t}}{5}

$$

Differential Equations

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turkey Diagram giving is we have a generator e and a resistor r with two arms resistance and in dr within doctrines off to Henry's and they're connected in cities and we have to find current at time T Another condition that is given to us is at time T equals to zero. Uh, current is one MP here and also generate a voltage is 40 times signs 60 t okay. And we have to deal graph that current as well. Okay, To solve this, we need to first convert this equation in tow. Sorry. Can work this graph into equation. So we'll care. Chops law, it shows law. And that says the total drop in voltage across the circuit is equals toe the vaulted supply. Therefore, we'll have easy equals toe drop across the resistor will be awry and drop across in dr will be l. D. I buy deity on substituting values is 40 sign 60 be artist 20 I plus Ellis to Henry's. So that is two times d I buy DP. Okay, now if you write this in standard form, we'll have two times the IBD T plus 20 I is equals to 40 signs 60 t Now we can divide by two so we'll have the a b d t plus 10. I is 20 divided by two so 20 times sign 60 t. Okay, now, if you look at the situation, this is the standard off standard form off linear equation that is divided by D X plus B y equals two Q. So we can calculate integrating factor as a race to integration off P D. X where Peace 10. So erase to integration off 10 D x, which is equals to integration off. Sorry, it is to 10 x right, 10 constant integration of one is X So our solution will be there four eyes it is to 10 x equals two integration off Q times that is 20 sign 60 de times a raise to 10 x d x. Okay, here, twenties constant. So it will come out off integration and we're left with It is to 10 x times sign 60 t. Sorry. This is not okay. So here we don't have the exits. Integrating factor will be here is to integration off PDT. So that is integration off piece tense or 10 DT and that gives us a raise to 10 t So our solution will be eyes that this I times so high times Yeah. Ah, yes, which is it is 2 20 integration off Q I f so Q Is 20 sign 60 d times I f. It is 2 20 DT. Okay, so we can take out 20 since it's a constant value and we'll have integration off sign 60 t it is 2. 20 deity. Okay, so now we can use a formula off integration. So that is integration off areas two x sign b x plus e d x. And it's answer is it is two x or a square plus B square and it's eight times sign Be explicit. See, minus b times cause B X plus e Yeah. Okay, so applying this formula here, we'll have 20 times. It is 2. 20 or what? 60 we have 60. So 60 square is 00 36 Bless 10 square. So that is 100. Then into eight times signed, we explore Asi So is 10 time sign we explosive. That is nothing About 60 de minus b times will be 60 so 60 times cause 60 d Okay, so this is a I plus c. So this is high times here is to 20. Now, if you simplify this, we get 20 over 3700. It is 2. 20. Here it is. 10 times sign 60 T minus 60 times cause 60 t plus c. Okay, so on Cal. Okay, so these are solution for areas to for general solution off, so we can simplify this a little. So this is I times it is 2. 20 equals two. It can cancel this and we can take out 10 from years. This will again become 20/3, 70 years to 20 on this becomes sign 60 be minus six times cause 60 d plus c. Okay, so we can divide by this value since we need value off I So I will be 20 so we can cancel against to over 37. It is 2 20 divided by it is 2. 20. This is science 60 will copy it as it is minus six times cause 60 80 plus c times. If it is divided, it will go in numerator by minus signing power. So that's absolution for I Now we need value off, See to get a particular solution tow. Fourth c were given a T equals to zero eyes, one mbia. Therefore, if you substitute I as one, this two by 37 it is 20 over. We can cancel these two terms so we don't have to write. So this becomes sign 60 times zero minus six times cause zero plus series to it is +20 Okay, so this is I sorry. One because 22 by 37 sign 0 60 time 00 So sign 00 minus called zero is one. So this is minus six plus c. So therefore, value off, see is can write it as one minus too bye. 37 minus six. So it is one plus two. Well, by 37. So on solving this, we get 37 plus 12. So that is 49/37. So that's a value off. See? So our solution is Therefore I particular will be to over 37. Sign 60 t minus six times cause 60 t class to buy. Sorry, it's not too. It's see value. That is 49 by 37 into areas to minus 20. Okay, so this function is two parts the district no metric and exponential, so starting the T value is less the exponential have greater role to play. And we'll have, let's say, a graph like this. But after a time this exponential esti grows, the exponential value becomes very small, enhance This value can be neglected and therefore initially function will start from 49 by 37 will come down and structure weight and then it will settle to a constant amplitude, okay?