solve the following Cauchy´s problem
Solve the following Cauchy problems under the given initial conditions. - - 1. -Uxx + Uz + (2 – sin(x) – cos (x))uy – (3 + cos²(x))uyy = 0 if the initial conditions is u(x, cox(x)) = 0, uz(x, c
Answers
The solution of the given partial differential equation is given by; $$ U(x,y,z) = \(-\frac{1}{2} e^{-\frac{1}{2}(y + z + \frac{sin(x) - cos(x)}{2})^2} - \frac{1}{2} e^{-\frac{1}{2}(y + z - \frac{sin(x) + cos(x)}{2})^2} \$\$\)
Given Cauchy's problem is; \(\$\$ -U_{xx} + U_z + (2 - sin(x) -cos(x))U_y - (3 + cos^2(x))U_{yy} = 0 \$\$\)
Initial condition is $u(x,0) = 0, \(u_z(x,0) = -e^{-x^2}\$\)
The general solution of the given partial differential equation is given by;
\(\$\$ U(x,y,z) = F(y + z + \frac{sin(x)}{2} - \frac{cos(x)}{2}) + G(y + z - \frac{sin(x)}{2} + \frac{cos(x)}{2}) \$\$\)
Where $F$ and $G$ are arbitrary functions of their arguments.
Now, applying the initial condition, we get; $$ \begin{aligned}
\(U(x,0,z) &= F(z + \frac{sin(x)}{2} - \frac{cos(x)}{2}) + G(z - \frac{sin(x)}{2} + \frac{cos(x)}{2}) = 0\)
\(U_z(x,0,z) &= F'(z + \frac{sin(x)}{2} - \frac{cos(x)}{2}) + G'(z - \frac{sin(x)}{2} + \frac{cos(x)}{2}) = -e^{-x^2}\) \end{aligned}$$
Now, we need to solve for $F$ and $G$ using the above conditions.
Solving for $F$ and $G$, we get;
\(\$\$ F(y + z + \frac{sin(x)}{2} - \frac{cos(x)}{2}) = -\frac{1}{2} e^{-\frac{1}{2}(z + y + \frac{cos(x)}{2} - \frac{sin(x)}{2})^2} \$\$\)
and \(\$\$ G(y + z - \frac{sin(x)}{2} + \frac{cos(x)}{2}) = -\frac{1}{2} e^{-\frac{1}{2}(z + y - \frac{cos(x)}{2} + \frac{sin(x)}{2})^2} \$\$\)
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Related Questions
3 cards are chosen at random from a standard 52-card deck. What is the probability that they form a pair
Answers
The probability of drawing a pair when selecting three cards at random from a standard 52-card deck is approximately 0.1694, or about 16.94%.
To calculate the probability of drawing a pair of cards from a standard 52-card deck when selecting three cards at random, we need to consider the number of favorable outcomes (pairs) and the total number of possible outcomes.
First, let's determine the number of favorable outcomes. For a pair, we need two cards of the same rank (e.g., two 7s, two Queens, etc.) and any third card that is different from the pair.
The number of ways to choose a rank for the pair is 13 (one for each rank). For each rank, there are four cards of that rank in the deck. Therefore, the number of ways to select two cards of the same rank is given by:
Number of ways to choose a pair = 13 * (4 choose 2) = 13 * 6 = 78
After selecting the pair, there are 50 cards remaining in the deck. The third card must be of a different rank from the pair. There are 48 cards remaining with different ranks from the pair.
Therefore, the number of favorable outcomes (pairs) is 78 * 48 = 3,744.
Next, let's calculate the total number of possible outcomes when selecting three cards from a standard deck. The total number of ways to choose any three cards from 52 cards is given by:
Total number of possible outcomes = (52 choose 3) = 22,100
Finally, we can calculate the probability of drawing a pair:
Probability of drawing a pair = Number of favorable outcomes / Total number of possible outcomes
= 3,744 / 22,100
≈ 0.1694
Therefore, the probability of drawing a pair when selecting three cards at random from a standard 52-card deck is approximately 0.1694, or about 16.94%.
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Find the Midpoint of the segment with the following endpoints (2,10) and (8,4)
Answers
Answer:
midpoint = (5,7)
Step-by-step explanation:
Use the midpoint equations. x1 + x2 / 2 and y1 + y2 / 2
2 + 8 = 10/2 = 5
10 + 4 = 14/2 = 7
Answer:
(4,5)
Step-by-step explanation:
Leave a like
only 10-11 please, giving brainliest!
Answers
Y +8
1/1
11) X + 2
Y + 2
1
I don’t know how u want the answer
Quantitative or qualitative
help pls I need this ASAP
Answers
Answer:
1. Quantitative data
2. Quantitative data
3. Qualitative data
4. Qualitative data
5. Quantitative data
6. Qualitative data
7. Quantitative data
8. Qualitative data
9. Quantitative data
10. Quantitative data
Step-by-step explanation:
Qualitative data usually answers the question "what kind". Quantitative data answers the question "how many" (or "how much" ).
Answer:
quantitative or qualitative
Prove the following trigonometric identities
csc x - sin x - cosx cos x
Answers
To prove the trigonometric identity csc(x) - sin(x) - cos(x)cos(x), we will start by simplifying the left-hand side of the equation using trigonometric definitions and identities.
Recall that csc(x) = 1/sin(x). We will use this definition to rewrite the left-hand side of the equation:
1/sin(x) - sin(x) - cos(x)cos(x)
Now, we will find a common denominator for the terms in the equation. In this case, the common denominator is sin(x). To do this, we will multiply sin(x) to the second term:
(1 - sin^2(x) - cos(x)cos(x)sin(x)) / sin(x)
Next, we will use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the expression:
(1 - (1 - cos^2(x)) - cos(x)cos(x)sin(x)) / sin(x)
Simplifying the expression, we get:
(cos^2(x) - cos(x)cos(x)sin(x)) / sin(x)
Now, we can factor out cos(x) from the numerator:
cos(x)(cos(x) - sin(x)) / sin(x)
This expression is equivalent to the given identity, so we have proven the trigonometric identity:
csc(x) - sin(x) - cos(x)cos(x) = cos(x)(cos(x) - sin(x)) / sin(x)
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Help this is due at 11:59
Answers
Answer:
y = 2x - 2
Step-by-step explanation: Can I get brainliest? Thanks!
1. Slope-intercept form: y=mx+b
m = slope
x = degree of slope
b = y-intercept
2. Solve
m = 4/2 = 2
b = -2
y = 2x - 2
Each square is worth is..
Answers
Answer:
10
Step-by-step explanation:
There are 100 squares in this square, so each tiny square is 10 because you take the square root of 100.
Hope this helps! :)
What is the value of the expression -218 - 72 - (-5)?
Answers
Answer:
The answer is -285
Step-by-step explanation:
x plus 2.7 is greater than or equal to 9.4
Answers
Answer:
X >= 6.7
Step-by-step explanation:
So first of all, what you need is write out the equation from the given prompt
x + 2.7 >= 9.4
Next, what you really want to do is
Subtract 2.7 both side
x >= 9.4 - 2.7
x >= 6.7
PLEASE ANSWER - BRAINLIEST IF 2 ANSWERS - 5 STAR RATING - THANKS (PROFILE TOO)
Five students plot their arm span and height
on the graph below.
Which of the following describes one
of these 5 students?
Answers
Answer:
She will save 1.24
Step-by-step explanation:
2.65 x 4 10 .60 plus .53 x 4 equals 12.72
6.48 x 2 equals 13.96
tell me if im wrong<3
Answer:
The correct answer would be D
Step-by-step explanation:
Most of the students have an average height around 166-180
and the arm span of 162-165
Student D has over the average 195 and an arm span of 188 so the correct answer would be D
(4x+5)(4x−5)
I'm stuck can someone please help me
Answers
Answer:6small2−25
Step-by-step explanation:
Answer:
16x² - 25
Step-by-step explanation:
(4x+5)(4x−5)
16x² - 20x + 20x - 25
16x² - 25
Is a function or non function explain
Answers
Answer:
It is a non function.
Step-by-step explanation:
Numbers cannot repeat in functions
The variable, c, represents Central's score before the three-pointer. Express the total points scored in the game as a variable expression. Check all that apply.
Answers
Corrected Question
Central High School plays Eastern High School in a basketball game. Eastern had double the score of Central before Central scored a three-pointer as the game ended.
The variable, c, represents Central's score before the three-pointer. Express the total points scored in the game as a variable expression. Check all that apply.
2c + c 2c + 3 3c + 3 2c + c – 3 2c – c + 3 2c + c + 3Answer:
(C)3c + 3
(F)2c + c + 3
Step-by-step explanation:
Central High School's Score = cEastern had double the score of Central, therefore Eastern High School's Score =2cCentral then scored a 3 pointer.Therefore:
Total Points Scored in the game =c+2c+3 (This is option F)
Simplifying, we have:
c+2c+3=3c+3 (This is option C)
Therefore, C and F are the correct options.
The box plots display data collected when two teachers asked their classes how many pencils they lose in a school year.
A box plot uses a number line from 5 to 47 with tick marks every one unit. The box extends from 8 to 14 on the number line. A line in the box is at 11. The lines outside the box end at 7 and 45. The graph is titled Mr. Johnson's Class, and the line is labeled Number Of Pencils.
A box plot uses a number line from 0 to 51 with tick marks every one unit. The box extends from 12 to 21 on the number line. A line in the box is at 14.5. The lines outside the box end at 0 and 50. The graph is titled Mr. Simpson's Class, and the line is labeled Number Of Pencils.
Which class lost the most pencils overall based on the data displayed?
Mr. Simpson's class; it has a larger median value 14.5 pencils
Mr. Johnson's class; it has a larger median of 11 pencils
Mr. Simpson's class; it has a narrow spread in the data
Mr. Johnson's class; it has a wide spread in the data
Answers
The class that lost the most pencils overall based on the data displayed is D. Mr. Johnson's class; it has a wide spread in the data
How to explain the informationThe answer is Mr. Johnson's class. The median is the middle value in a set of data. In Mr. Johnson's class, the median is 11 pencils. This means that half of the students in his class lost 11 or fewer pencils, and half of the students lost 11 or more pencils.
In Mr. Simpson's class, the median is 14.5 pencils. This means that half of the students in his class lost 14.5 or fewer pencils, and half of the students lost 14.5 or more pencils.
Since the median for Mr. Johnson's class is lower than the median for Mr. Simpson's class, we can conclude that Mr. Johnson's class lost more pencils overall.
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A skin patch contains a new drug to help people quit smoking. A group of 75 cigarette smokers have volunteered as subjects to test the new skin patch. For one month, 40 of the volunteers receive skin patches with the new drug. The other volunteers receive skin patches with no drugs. At the end of the two months, each subject is surveyed regarding his or her current smoking habits.
a) Is this an experiment or an observational study?
b) Identify the control group, if any, If no control group, write NONE.
c) Identify the treatment group, if any, If no treatment group, write NONE.
d) Is there a placebo being used or not? Explain.
Answers
A) The given problem can be considered as an experiment as the researcher is trying to observe the effect of a new drug on quitting smoking.
B) In this experiment, the control group is composed of the participants who received skin patches with no drugs. They are used for the comparison of the effects of the treatment.
C) The treatment group in this experiment is composed of the participants who received skin patches with the new drug. This group is used to analyze the effect of the treatment.
D) Yes, there is a placebo being used in this experiment, which is the skin patch without the new drug. It is used to evaluate the effectiveness of the new drug.
Explanation:
An experiment is a study in which the researcher changes one or more variables and observes or measures the effect of those changes on another variable. In this given problem, the researcher is trying to observe the effect of a new drug on quitting smoking. Thus, it can be considered an experiment.
The control group is a group of participants who do not receive the treatment and are used for comparison with the treatment group. In this experiment, the participants who received skin patches with no drugs acted as the control group.
The treatment group, on the other hand, is a group of participants who receive the treatment and are used to analyze the effect of the treatment. In this experiment, the participants who received skin patches with the new drug acted as the treatment group.
A placebo is an inactive substance that has no therapeutic effect but is used in studies to evaluate the effectiveness of a drug or intervention. In this experiment, the skin patch without the new drug is used as a placebo to evaluate the effectiveness of the new drug. Thus, it is being used in the given experiment.
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with a mean of 34 calls per hour. The service rate per line is 18 calls per hour. (a) What is the probability that \( 0,1,2 \), and 3 access lines will be in use? (Round your answers to four decimal p
Answers
The probabilities of having 0, 1, 2, and 3 access lines in use are approximately 0.000048, 0.001636, 0.011072, and 0.047368, respectively. These probabilities are calculated using the Poisson distribution formula based on the given mean arrival rate of 34 calls per hour and a service rate of 18 calls per hour.
To determine the probabilities of having 0, 1, 2, and 3 access lines in use, we can use the Poisson distribution formula. Given a mean arrival rate of 34 calls per hour and a service rate of 18 calls per hour, we can calculate the probabilities as follows:
For 0 access lines in use, the probability can be calculated using the formula \(P(X = 0) = \frac {e^{-\lambda} * \lambda^0}{0!}\) where λ is the mean arrival rate. Substituting the values, we have \(P(X = 0) = e^{-34} * (34^0) / 0! = 0.000048\).
Similarly, for 1, 2, and 3 access lines in use, we can calculate the probabilities using the same formula. The probabilities are:
\(P(X = 1) = e^{-34} * (34^1) / 1! = 0.001636\)
\(P(X = 2) = e^{-34} * (34^2) / 2! = 0.011072\)
\(P(X = 3) = e^{-34} * (34^3) / 3! = 0.047368\)
Therefore, the probabilities of having 0, 1, 2, and 3 access lines in use are approximately 0.000048, 0.001636, 0.011072, and 0.047368, respectively.
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If two samples are selected from the same population, under what circumstances will the two samples have exactly the same t statistic?
Answers
If two samples are selected from the same population, the two samples have exactly the same t statistics if the samples are same size and have the same variance and mean
The t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error.
Variance is the expectation of the squared deviation of a random variable from its population mean or sample mean.
Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers
Hence, If two samples are selected from the same population, the two samples have exactly the same t statistics if the samples are same size and have the same variance and mean
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Stella sold five items at a
garage sale for $12.50, $2.75,
$0.50, $20.00, and $8.50.
How much money did she
make? Plz help
Answers
Answer:
$44.25
Step-by-step explanation:
12.5 + 2.75 + .5 + 20 + 8.5 =44.5
2.75 + .5 = 3.25
20 + 12.5 + 32.5
3.25 + 32.5 + 8.5 = 44.25
what is the correlation
Answers
Answer: moderate, C, B
Step-by-step explanation:
A) A shows a moderate negative correlation. It is moderate because the scattered points are sort of close to the line so it has moderate/medium correlation. It is also negative because it has a negative slope
B) C shows the strongest correlation because the points around the line are tight and close.
C) B should not have been drawn. The correlation is very weak. You do know where the line should be because the points are all over the place.
The dimensions of a rectangular prism are 5 cm, 5 cm, and 2 cm.
Determine the surface area.
A. 50 cm?
B. 25 cm2
C. 60 cm2
D. 120 cm2
E. 90 cm2
ANSWER QUICK PLEASE
Answers
As the earth rotates, it travels a distance of 80 miles in 5 minutes. At this constant rate, which equation best represents y, the total distance the earth travels in x minutes?
Answers
Answer:
y=16x
Step-by-step explanation:
So, you are given the equation as if y=total distance the earth travels in x minutes. If you are given that you travel 80 miles in 5 minutes, you can plug these value into the variables they are representing and solve for the constant.
Visual:
y=xk y=miles x=minutes k=constant
Plug in your values and you get:
80=5k
Now, you solve for k by diving 5 on both sides, which leaves you with k=16.
Now you plug your value for k into the original equation, leaving you with y=16x
Please help me I need this to be absolutely correct thanks!
Answers
Answer:
Step-by-step explanation:
I can't really graph it for you, but I will show you how to.
-x is the value on the horizontal axis, which are 1,2,3,4,6
if you replace x for 1, which is
y= -4(1) + 3
y= -1
so you can graph that point (1,-1)
you can replace x with other numbers on the horizontal axis.
Hope this helps
Find the value of k if the graph of y=kx passes through the given point.
A(4,-80)
K=?
Answers
The value of k from the equation is k = -20
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
y = kx be equation (1)
Let the first point be P ( 4 , -80 )
Substituting the values in the equation , we get
-80 = 4k
Divide by 4 on both sides of the equation , we get
k = - ( 80/4 )
On simplifying the equation , we get
k = -20
Hence , the equation is k = -20
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Please help me pretty please
Answers
Answer:
28.6
Step-by-step explanation:
Pathagorem therom
Use matlab to construct the augmented matrix [a b] and then perform row reduction using the rref() function. write out your reduced matrix and identify the free and basic variables of the system
Answers
The augmented matrix [a b] is constructed in MATLAB, and row reduction is performed using the rref() function. The reduced matrix is obtained, and the free and basic variables of the system are identified.
To construct the augmented matrix [a b] in MATLAB, we can define a matrix with two columns containing the values of variables a and b. For example, if we have a = 2 and b = 3, the augmented matrix can be constructed as follows:
```
A = [2 3];
```
Next, we can use the rref() function in MATLAB to perform row reduction on the augmented matrix. The rref() function returns the reduced row echelon form of the matrix. We can store the result in a new matrix, let's say R:
```
R = rref(A);
```
The reduced matrix R will contain the coefficients of the variables and the constants of the system. By analyzing the reduced matrix, we can determine the basic and free variables. Basic variables are those corresponding to the leading entries (pivots) in each row of the reduced matrix. Free variables, on the other hand, are associated with the columns that do not contain a pivot.
The reduced matrix R will have a row of zeros for any free variables. By observing which columns have pivots and which do not, we can identify the basic and free variables. The basic variables correspond to the columns with pivots, while the free variables are associated with the columns without pivots.
In summary, by constructing the augmented matrix [a b] and performing row reduction using the rref() function in MATLAB, we obtain a reduced matrix R. By analyzing the structure of R, we can identify the basic and free variables of the system.
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You move down 7 units. You end at (7, 3). Where did you start?
How do I solve it

Answers
Answer: (7,10)
Step-by-step explanation:
all the students in an algebra class took a 100-point test. five students scored 100, each student scored at least 60, and the mean score was 76. what is the smallest possible number of students in the class?
Answers
The smallest possible number of students in the algebra class is 13.
What is meant by the word inequality?Inequalities define the connection between two non-equal values. Inequality means being unequal. If two values really aren't equal, we use the "not equal symbol (≠)".Let n≥5 be the number of students.
The sum of their scores must be at least 5×100 + (n - 5).
Simultaneously, we must achieve the mean 76, which is equivalent to achieving the mean sum 76n.
As a result, we have a sufficient precondition on n: we should have 5×100 + (n - 5) ≤ 76n.
This can be reduced to 200 ≤ 16n. This is true for the smallest integer n = 13.
To complete our solution, we must now determine how 13 students might have scored just on test.
We have 13×76 = 988 points to distribute to them. Because five 100s equal 500, we must distribute the remaining 488 points as among remaining eight students.
This can be accomplished, for example, by awarding each of them 61 points.
Thus, the smallest possible number of students in the class is 13.
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Consider the initial value problem y’ = ty^3, y(0) = 1/5.
(a) Solve. (Using separation of variables).
(b) Does your solution make sense at t=6?
(c) If this equation was describing something like the energy of a system, what happens at t=5?
Answers
(a)the solution to the initial value problem is:
y =\(\frac{1}{\sqrt{\frac{1}{25}-\frac{t^{2} }{2}} }\)
(b) Since the argument of the square root is negative, y(6) is not defined. Therefore, the solution does not make sense at t=6.
(c)This means that the inverse of the energy of the system becomes infinite at t=5, which indicates a singularity or a breakdown of the model at that point. This could imply a physical phenomenon such as a phase transition or an instability in the system.
(a) We have the differential equation:
\(\frac{dy}{dt}\) = t\(y^{3}\)
Separating the variables, we get:
\(\frac{1}{y^{3} }\) dy = t dt
Integrating both sides, we get:
\(\frac{-1}{2y^{2} }\)= \(\frac{1}{2t^{2} }\) + C
where C is the constant of integration. To find C, we use the initial condition y(0) = \(\frac{1}{5}\):
\(\frac{-1}{2(\frac{1}{5} )^{2} }\)= \(\frac{1}{2(0)^{2} }\) + C
C = \(\frac{-1}{50}\)
Thus, the solution to the initial value problem is:
\(\frac{-1}{2y^{2} }\) =\(\frac{1}{2t^{2} }\) - \(\frac{1}{50}\)
Solving for y, we get:
y =\(\frac{1}{\sqrt{\frac{1}{25}-\frac{t^{2} }{2}} }\)
We need to check if y(6) is defined or not. We have:
y(6) = \(\frac{1}{\sqrt{\frac{1}{25}-\frac{6^{2} }{2} } }\) = \(\frac{1}{\sqrt{\frac{-34}{25} } }\)
Since the argument of the square root is negative, y(6) is not defined. Therefore, the solution does not make sense at t=6.
(c) If the equation was describing something like the energy of a system, the quantity y^2 would represent the inverse of the energy of the system. At t=5, we have:
y(5) = \(\frac{1}{\sqrt{\frac{1}{25}-\frac{5^{2} }{2} } }\) = 0
This means that the inverse of the energy of the system becomes infinite at t=5, which indicates a singularity or a breakdown of the model at that point. This could imply a physical phenomenon such as a phase transition or an instability in the system.
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Help me now please i want to finish this today so I can finish summer school
Answers
same i pass im at like 70% Step-by-step explanation:
slope intercept form of y+5=-7/4 ( x - 4)
Answers
Answer: y=-7/4x+2
Step-by-step explanation: