9+ k = 18 A. 18 B.9 C.8 D.7 Help please

y=c1e^x+c2e^−x is a two-parameter family of solutions of the second order differential equation y′′−y=0. Find a solution of the

vagabundo [1.1K]

The general form of a solution of the differential equation is already provided for us:

$y(x) = c_1 \textrm{e}^x + c_2\textrm{e}^{-x},$

where $c_1, c_2 \in \mathbb{R}$ . We now want to find a solution $y$ such that $y(-1)=3$ and $y'(-1)=-3$ . Therefore, all we need to do is find the constants $c_1$ and $c_2$ that satisfy the initial conditions. For the first condition, we have: $y(-1)=3 \iff c_1 \textrm{e}^{-1} + c_2 \textrm{e}^{-(-1)} = 3 \iff c_1\textrm{e}^{-1} + c_2\textrm{e} = 3.$

For the second condition, we need to find the derivative $y'$ first. In this case, we have:

$y'(x) = \left(c_1\textrm{e}^x + c_2\textrm{e}^{-x}\right)' = c_1\textrm{e}^x - c_2\textrm{e}^{-x}.$

Therefore:

$y'(-1) = -3 \iff c_1\textrm{e}^{-1} - c_2\textrm{e}^{-(-1)} = -3 \iff c_1\textrm{e}^{-1} - c_2\textrm{e} = -3.$

This means that we must solve the following system of equations:

$\begin{cases}c_1\textrm{e}^{-1} + c_2\textrm{e} = 3 \\ c_1\textrm{e}^{-1} - c_2\textrm{e} = -3\end{cases}.$

If we add the equations above, we get:

$\left(c_1\textrm{e}^{-1} + c_2\textrm{e}\right) + \left(c_1\textrm{e}^{-1} - c_2\textrm{e} \right) = 3-3 \iff 2c_1\textrm{e}^{-1} = 0 \iff c_1 = 0.$

If we now substitute $c_1 = 0$ into either of the equations in the system, we get:

$c_2 \textrm{e} = 3 \iff c_2 = \dfrac{3}{\textrm{e}} = 3\textrm{e}^{-1.}$

This means that the solution obeying the initial conditions is:

$\boxed{y(x) = 3\textrm{e}^{-1} \times \textrm{e}^{-x} = 3\textrm{e}^{-x-1}}.$

Indeed, we can see that:

$y(-1) = 3\textrm{e}^{-(-1) -1} = 3\textrm{e}^{1-1} = 3\textrm{e}^0 = 3$

$y'(x) =-3\textrm{e}^{-x-1} \implies y'(-1) = -3\textrm{e}^{-(-1)-1} = -3\textrm{e}^{1-1} = -3\textrm{e}^0 = -3,$

which do correspond to the desired initial conditions.

3 0

3 years ago

Find the first four terms of each sequence. F(n) = 3n - 1

timofeeve [1]

Answer:

2, 5, 8, 11

Step-by-step explanation:

F(n) = 3n - 1

put n=1

F(1) = 3(1) - 1 = 2

put n=2

F(2) = 3(2) - 1 = 6 - 1 = 5

put n=3

F(3) = 3(3) - 1 = 9 - 1 = 8

put n=4

F(4) = 3(4) - 1 = 12 - 1 = 11

5 0

3 years ago

Graph the equation y = 4/5x - 1

astra-53 [7]

Answer: The answer is the picture shown below

Step-by-step explanation:

To do this you will need to make a coordinate plane about 20 up and 20 down. So there is a formula that is y=mx+b which is called the slope intercept form and as you can see the problem given to you was already given in this formula. The variable m in the equation is the slope and slope is also known as rise (which basically means go up if positive and down if negative) over run (which is right if positive and left if negative) and as you can see in the equation the number 4/5 is the variable m. Since slope is rise over run 4 would be the rise and 5 would be the run, so you would go up 4 and right 5 because they are both positive. Now there is another variable that is b which would be the -1. In slope intercept formula b is the y intercept so -1 is where the line would cross the y axis. Now that you know all the parts to the slope you can put it together giving you the answer.

4 0

3 years ago

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Find the area of the circle with a circumference of 18.84 miles. Use 3.14 for π.

LenaWriter [7]

2.473 is the answer

7 0

3 years ago

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Please help, no links or you're getting reported and yourass is done

kvv77 [185]

What is there to answer! AH HA HA HA.

*smacks keyboard on head*

Please read this.

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The state of our economy is in a constant flux. Every single day, the stocks go up or go down or will stay the same. If elected president, I will ask the Federal Reserve to take a good long look at the interest rate and decide whether or not to change it. If elected president, I will create jobs where there are none, and where there are jobs, I will create internships.

And to those of you who would say, “I don’t think children are our future,” I must say in the strongest of terms: “I disagree.” Sometimes you have to take a stand for what you believe in like I do. We all should.

I will say our country is sharply divided over a war that is being waged in this land, all lands. My views on this war are clear: it is happening, it is happening in everywhere, and it will continue to happen until it is cured.

What do you mean by cured?

Well is a virus, specifically one called Coronavirus...

Some people believe we should withdraw all the activities we do now. Some people believe we do all the activities we want to right now. Some people believe we should stay and fight it until we’ve established a stable nation. Some people feel like we have handed over control to the virus. I feel we should fight it and develop a cure to live in a country with our deserved freedom, we cannot and will not hand it over to a virus.

We all need some socializing, not like you kind people are doing today but instead what activities you would prefer to do.

- I care deeply about social service programs and providing opportunities for low-income households

- provide free tuitions for doctors

- close the income gap and raise the minimum wage

- I appreciate everyone who wakes up every day and works hard for their community

Sound goods?

4 0

3 years ago

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