Both functions are the solution to the given Laplace solution.
Given Laplace's equation: ![u_{x x}+u_{y y}=0](https://tex.z-dn.net/?f=u_%7Bx%20x%7D%2Bu_%7By%20y%7D%3D0)
- We must determine whether a given function is the solution to a given Laplace equation.
- If a function is a solution to a given Laplace's equation, it satisfies the solution.
(1) ![u=e^{-x} \cos y-e^{-y} \cos x](https://tex.z-dn.net/?f=u%3De%5E%7B-x%7D%20%5Ccos%20y-e%5E%7B-y%7D%20%5Ccos%20x)
Differentiate with respect to x as follows:
![u_x=-e^{-x} \cos y+e^{-y} \sin x\\u_{x x}=e^{-x} \cos y+e^{-y} \cos x](https://tex.z-dn.net/?f=u_x%3D-e%5E%7B-x%7D%20%5Ccos%20y%2Be%5E%7B-y%7D%20%5Csin%20x%5C%5Cu_%7Bx%20x%7D%3De%5E%7B-x%7D%20%5Ccos%20y%2Be%5E%7B-y%7D%20%5Ccos%20x)
Differentiate with respect to y as follows:
![u_{x x}=e^{-x} \cos y+e^{-y} \cos x\\u_{y y}=-e^{-x} \cos y-e^{-y} \cos x](https://tex.z-dn.net/?f=u_%7Bx%20x%7D%3De%5E%7B-x%7D%20%5Ccos%20y%2Be%5E%7B-y%7D%20%5Ccos%20x%5C%5Cu_%7By%20y%7D%3D-e%5E%7B-x%7D%20%5Ccos%20y-e%5E%7B-y%7D%20%5Ccos%20x)
Supplement the values in the given Laplace equation.
![e^{-x} \cos y+e^{-y} \cos x-e^{-x} \cos y-e^{-y} \cos x=0](https://tex.z-dn.net/?f=e%5E%7B-x%7D%20%5Ccos%20y%2Be%5E%7B-y%7D%20%5Ccos%20x-e%5E%7B-x%7D%20%5Ccos%20y-e%5E%7B-y%7D%20%5Ccos%20x%3D0)
The given function in this case is the solution to the given Laplace equation.
(2) ![u=\sin x \cosh y+\cos x \sinh y](https://tex.z-dn.net/?f=u%3D%5Csin%20x%20%5Ccosh%20y%2B%5Ccos%20x%20%5Csinh%20y)
Differentiate with respect to x as follows:
![u_x=\cos x \cosh y-\sin x \sinh y\\u_{x x}=-\sin x \cosh y-\cos x \sinh y](https://tex.z-dn.net/?f=u_x%3D%5Ccos%20x%20%5Ccosh%20y-%5Csin%20x%20%5Csinh%20y%5C%5Cu_%7Bx%20x%7D%3D-%5Csin%20x%20%5Ccosh%20y-%5Ccos%20x%20%5Csinh%20y)
Differentiate with respect to y as follows:
![u_y=\sin x \sinh y+\cos x \cosh y\\u_{y y}=\sin x \cosh y+\cos x \sinh y](https://tex.z-dn.net/?f=u_y%3D%5Csin%20x%20%5Csinh%20y%2B%5Ccos%20x%20%5Ccosh%20y%5C%5Cu_%7By%20y%7D%3D%5Csin%20x%20%5Ccosh%20y%2B%5Ccos%20x%20%5Csinh%20y)
Substitute the values to obtain:
![-\sin x \cosh y-\cos x \sinh y+\sin x \cosh y+\cos x \sinh y=0](https://tex.z-dn.net/?f=-%5Csin%20x%20%5Ccosh%20y-%5Ccos%20x%20%5Csinh%20y%2B%5Csin%20x%20%5Ccosh%20y%2B%5Ccos%20x%20%5Csinh%20y%3D0)
The given function in this case is the solution to the given Laplace equation.
Therefore, both functions are the solution to the given Laplace solution.
Know more about Laplace's equation here:
brainly.com/question/14040033
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The correct question is given below:
Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.) u = e^(−x) cos(y) − e^(−y) cos(x) u = sin(x) cosh(y) + cos(x) sinh(y)
![9:7](https://tex.z-dn.net/?f=9%3A7)
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Answer:
(0, -2)
Step-by-step explanation:
You find an ordered pair by making x = 0, and solving for y, like so:
2x - y = 2
2(0) - y = 2
-y = 2 → Divide both sides by -1
y = -2
<u>Now your ordered pair for this would be (0, -2)</u>
Answer:
20%
Step-by-step explanation:
250*4=1000
200*4=800
1000-800=200
200 is 1/5 of 1000
1/5=20%
Answer:
y = 3x
Step-by-step explanation:
x is number of cartons of eggs
y is cost
cost increases by 3 for every carton of eggs, so slope is 3
x and y intercepts are zero because there is no cost when there are no eggs
equation: y = 3x