The strength of an electrical current x flowing through the electric circuit shown is expressed as a function of time t and satisfies the following differential equation:
[tex]\displaystyle \large{L \frac{dx}{dt} + Rx = V}[/tex]
Find the strength of the electrical current x after switch S is closed at time t = 0. Assume that L, R and V are positive constants, and also that x = 0 when t = 0. Then, find [tex]\displaystyle \large{ \lim_{t \to \infty} x}[/tex]
Topic: Application of Differential Equation Reviews

Answer:

The current of the circuit at t = 0 is equal to 0.

If we take the limit as t approaches infinity, the current is equal to ε/R or V/R.

General Formulas and Concepts:

Calculus

Differentiation

Derivative Property [Multiplied Constant]:

[tex]\displaystyle (cu)' = cu'[/tex]

Derivative Property [Addition/Subtraction]:

[tex]\displaystyle (u + v)' = u' + v'[/tex]

Derivative Rule [Basic Power Rule]:

Slope Fields

Integration

Integration Rule [Reverse Power Rule]:

[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:

[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:

[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Method: U-Substitution

Electricity

Ohm's Law: V = IR

Circuits

Step-by-step explanation:

*Note:

In the given equation, our variable of differentiation is x. I will rewrite this as current I for physics notation purposes.

Step 1: Define

Identify given.

[tex]\displaystyle L \frac{dI}{dt} + RI = V[/tex]

[Assuming switch S is closed] Recall that an inductor is used in a circuit to resist change. After a long period of time, when it hits steady-state equilibrium, we expect to see the inductor act like a wire.

Step 2: Find Current Expression Pt. 1

Step 3: Find Current Expression Pt. 2

Identify variables for u-substitution.

Step 4: Find Current Expression Pt. 3

Recall that our initial condition is when t = 0, denoted as u₀, and we go to whatever position u we are trying to find. Also recall that time t always ranges from t = 0 (time can't be negative) and to whatever t we are trying to find.

Recall that our initial condition u₀ (derived from Ohm's Law) contains only the voltage across resistor R, where voltage is supplied by the given battery. This is because the current is stopped once it reaches the inductor in the circuit since it resists change.

Step 5: Solve

If we are trying to find the strength of the electrical current I at t = 0, we simply substitute t = 0 into our current function:

[tex]\displaystyle\begin{aligned}I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}t} \\I(0) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}(0)} \\& = \boxed{\bold{0}}\end{aligned}[/tex]

If we are taking the limit as t approaches infinity of the current function I(t), we are simply just trying to find the current after a long period of time, which then would just be steady-state equilibrium:

[tex]\displaystyle\begin{aligned}I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}t} \\\lim_{t \to \infty} I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}(\infty)} \\& = \boxed{\bold{\frac{\mathcal E}{R}}}\end{aligned}[/tex]

∴ we have found the current I at t = 0 and the current I after a long period of time and proved that an inductor resists current running through it in the beginning and acts like a wire when in electrical equilibrium.

---

Topic: AP Physics C - EMAG

Unit: Induction

Round each fraction to help you estimate the solution for the following equation: (2 points) nine tenths minus eight tenths equals a 0 b one half c 1 d one and one half

Determine the volume of the cylinder. Use 3.14 for T.
2 in.
SOUP
6.5 in.
cubic inches

Please Answer Both Questions!
If your correct I'll mark you brainliest if someone else answers


Only need answers

Look at △ABC and △DEF on this coordinate grid.



Which transformation shows that △ABC and △DEF are similar?

A
a dilation of △ABC with a scale factor of 2

B
a translation of △ABC 1 unit up and 3 units to the right

C
a dilation of △ABC with a scale factor of 12

D
a translation of △ABC 3 units up and 3 units to the right

If the relative frequencies being compared are 0.21 and 0.79, which conclusion is most likely supported by the data? an association cannot be determined between the categorical variables because the relative frequencies are not similar in value. there is likely an association between the categorical variables because the relative frequencies are not similar in value. an association cannot be determined between the categorical variables because the sum of the relative frequencies is 1.0. there is likely an association between the categorical variables because the sum of the relative frequencies is 1.0.

Lin needs `5` cups of flour to make muffins for her family, but only has `3. 5` cups of flour left. How much more flour does she need?

Good evening! Can someone please answer this, ill give you brainliest and your earning 50 points. Would be very appreciated.

The sum of two numbers is 53 The larger number is 11 more than the smaller number. What are the numbers?

an electrician completes 1/6 of a job in 2/3 hour. at this rate, how long does it take the electrician to complete the job?


PLEASE HELP!!!! IM GONNA GET IN TROUBLE

13) Which expression is equivalent to 2(5 – 3x)?
-
A) 10 - 5x
B) 10 - 6x
-
07 - 5x
D) 7 6x

Complete the equivalent equation for –7x – 60 = x2 10x. (x )(x ) = 0 what are the solutions of –7x – 60 = x2 10x? x =

Selina is picking strawberries. She is using 5 boxes to hold them all. Each box has a width of 6 in, a height of 3 in, and a length of 4 in. What is the total volume of all 5 boxes? Explain how to solve this problem.

Are these ratios equivalent? 4:5 12:20

Mr. Lee has 61 pencils. He gave some pencils away. Now he has 33 pencils left. Which equation can be used to find how many pencils Mr. Lee gives away.

What are the solutions of x2 = 8 – 5x? startfraction 5 minus startroot 57 endroot over 2 endfraction comma startfraction 5 startroot 57 endroot over 2 endfraction startfraction negative 5 minus startroot 57 endroot over 2 endfraction comma startfraction negative 5 startroot 57 endroot over 2 endfraction startfraction negative 47 over 4 endfraction comma startfraction 67 over 4 endfraction startfraction negative 67 over 4 endfraction comma startfraction 47 over 4 endfraction

...................................................................

A hot air balloon is 1,200 feet above the ground. The angle of depression from the basket of
the hot air balloon to the base of a monument is 54°.

Which equation can be used to determine the distance, d, in feet, from the basket of the hot
air balloon to the base of the monument?

Thirty
five years ago, the average price of a textbook was $36. Today, the average textbook is $79. What percent increase is this? Round to the nearest percent if needed
A.54%
B.67%
C.119%
D.107%

pls help its an exam

The table below represents the data collected at a sandwich shop for the last six months with respect to the type of bread people ordered (sourdough or wheat) and whether or not they got cheese on their sandwich.

What is the P(cheese | wheat) ? Show all work to receive full credit.

Explain how to graph the point (-4, 6)

I already know what a and b are i just need help on c

11 is a quotient of a number y and 6

Which is f(6) for the quadratic function graphed?
-2
-0.5
1.5
ОО

PLEASE HELP!!!!!!!!!!!!!!!! what is the perimeter of the trapezoid?

How many 10 inch pieces of bar can be cut from a bar that measures 120 inches?
Please explain !!​

Solve the formula S = 47tr2 for r.

Solve for x.
Your answer must be simplified.

1. Ms. Rodriquez wanted to grow out her hair. She measured the growth each month. The first month
her hair grew 1.775 centimeters. The next month it grew 1.45 centimeters. The third month it grew
1.2 centimeters. How many centimeters did Ms. Rodriquez's hair grow in three months?

An inground swimming pool contains 2.3 times 10 superscript 4 units of water. which unit of measure was used? teaspoons cubic centimeters fluid ounces gallons

Figure b 32.5 12.5 30 figure a 13 12 5