y = y_h + y_p = c1e^(6x) + c2e^(-6x) + (-1/12)x - 1/36 + (1/36)e^x.This is the solution to the given differential equation using the Method of Undetermined Coefficients.
To solve the given differential equation, y" - 36y = 3x + e, using the Method of Undetermined Coefficients, we first consider the homogeneous solution. The characteristic equation is r^2 - 36 = 0, which gives us the roots r1 = 6 and r2 = -6. Therefore, the homogeneous solution is y_h = c1e^(6x) + c2e^(-6x), where c1 and c2 are constants.
Next, we focus on finding the particular solution for the non-homogeneous term. Since we have a linear term and an exponential term on the right-hand side, we assume a particular solution of the form y_p = Ax + B + Ce^x.
Differentiating y_p twice, we find y_p" = 0 + 0 + Ce^x = Ce^x, and substitute into the original equation:
Ce^x - 36(Ax + B + Ce^x) = 3x + e
Simplifying the equation, we have:
(C - 36C)e^x - 36Ax - 36B = 3x + e
Comparing the coefficients, we find C - 36C = 0, -36A = 3, and -36B = 1.
Solving these equations, we get A = -1/12, B = -1/36, and C = 1/36.
Therefore, the particular solution is y_p = (-1/12)x - 1/36 + (1/36)e^x.
Finally, the general solution is the sum of the homogeneous and particular solutions:
y = y_h + y_p = c1e^(6x) + c2e^(-6x) + (-1/12)x - 1/36 + (1/36)e^x.
This is the solution to the given differential equation using the Method of Undetermined Coefficients.
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Identify those below that are linear PDEs. 8²T (a) --47=(x-2y)² (b) Tªrar -2x+3y=0 ex by 38²T_8²T (c) -+3 sin(7)=0 ay - sin(y 2 ) = 0 + -27+x-3y=0 (2)
Linear partial differential equations (PDEs) are those in which the dependent variable and its derivatives appear linearly. Based on the given options, the linear PDEs can be identified as follows:
(a) -47 = (x - 2y)² - This equation is not a linear PDE because the dependent variable T is squared.
(b) -2x + 3y = 0 - This equation is a linear PDE because the dependent variables x and y appear linearly.
(c) -27 + x - 3y = 0 - This equation is a linear PDE because the dependent variables x and y appear linearly.
Therefore, options (b) and (c) are linear PDEs.
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Consider the following statement about three sets A, B and C: If A n (B U C) = Ø, then A n B = Ø and A n C = 0.
Find the contrapositive and converse and determine if it's true or false, giving reasons. Finally, determine if the original statement is true.
The original statement is: If A n (B U C) = Ø, then A n B = Ø and A n C = Ø.1. Contrapositive: The contrapositive of the original statement is: If A n B ≠ Ø or A n C ≠ Ø, then A n (B U C) ≠ Ø.
2. Converse: The converse of the original statement is: If A n B = Ø and A n C = Ø, then A n (B U C) = Ø.
Now let's analyze the contrapositive and converse statements:
Contrapositive:
The contrapositive statement states that if A n B is not empty or A n C is not empty, then A n (B U C) is not empty. This statement is true. If A has elements in common with either B or C (or both), then those common elements will also be in the union of B and C. Therefore, the intersection of A with the union of B and C will not be empty.
Converse:
The converse statement states that if A n B is empty and A n C is empty, then A n (B U C) is empty. This statement is also true. If A does not have any elements in common with both B and C, then there will be no elements in the intersection of A with the union of B and C.
Based on the truth of the contrapositive and converse statements, we can conclude that the original statement is true.
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A person must score in the upper 5% of the population on an IQ test to qualify for a particular occupation.
If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for this occupation?
working please
A person must have an IQ score of approximately 124.68 or higher to qualify for this occupation.
We have,
To determine the IQ score that corresponds to the upper 5% of the population, we need to find the z-score that corresponds to the desired percentile and then convert it back to the IQ score using the mean and standard deviation.
Given:
Mean (μ) = 100
Standard deviation (σ) = 15
Desired percentile = 5%
To find the z-score corresponding to the upper 5% of the population, we look up the z-score from the standard normal distribution table or use a calculator.
The z-score corresponding to the upper 5% (or the lower 95%) is approximately 1.645.
Once we have the z-score, we can use the formula:
z = (X - μ) / σ
Rearranging the formula to solve for X (IQ score):
X = z x σ + μ
Substituting the values:
X = 1.645 x 15 + 100
Calculating the result:
X = 24.675 + 100
X ≈ 124.68
Therefore,
A person must have an IQ score of approximately 124.68 or higher to qualify for this occupation.
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let f be a function that is continuous on the closed interval 2 4 with f(2)=10 and f(4)=20
There exists a value c in the interval (2, 4) such that f(c) = 15.
Given that f is a function that is continuous on the closed interval [2, 4] and f(2) = 10 and f(4) = 20, we can use the Intermediate Value Theorem to show that there exists a value c in the interval (2, 4) such that f(c) = 15.
The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b], and if M is any value between f(a) and f(b) (inclusive), then there exists at least one value c in the interval (a, b) such that f(c) = M.
In this case, f(2) = 10 and f(4) = 20, and we are interested in finding a value c such that f(c) = 15, which is between f(2) and f(4). Since f is continuous on the interval [2, 4], the Intermediate Value Theorem guarantees that such a value c exists.
Therefore, there exists a value c in the interval (2, 4) such that f(c) = 15.
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6. Shawn (280 lbs) runs stairs for 45 minutes at a rate of 15 METs. What is his total caloric expenditure in kcals? 7. Sheryl (114 lbs) rode her motor scooter for 20 minutes to get to class (MET= 2.5). What was her total caloric expenditure for this activity?
1. Shawn's total caloric expenditure is 4,200 kcals.
2. Sheryl's total caloric expenditure is 190 kcals.
1. To calculate Shawn's total caloric expenditure, we can use the formula: Caloric Expenditure (kcal) = Weight (lbs) × METs × Duration (hours). Given that Shawn weighs 280 lbs, runs stairs at a rate of 15 METs, and exercises for 45 minutes (which is equivalent to 0.75 hours), we can substitute these values into the formula:
Caloric Expenditure = 280 lbs × 15 METs × 0.75 hours = 4,200 kcals
Therefore, Shawn's total caloric expenditure is 4,200 kcals.
2. Similarly, to calculate Sheryl's total caloric expenditure, we use the same formula: Caloric Expenditure (kcal) = Weight (lbs) × METs × Duration (hours). Given that Sheryl weighs 114 lbs, rides her motor scooter with a MET value of 2.5, and rides for 20 minutes (which is equivalent to 0.33 hours), we can substitute these values into the formula:
Caloric Expenditure = 114 lbs × 2.5 METs × 0.33 hours = 190 kcals
Therefore, Sheryl's total caloric expenditure for riding her motor scooter is 190 kcals.
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Solve the difference equation
Xt+1 = 0.99xt - 4, t = 0, 1, 2, ...,
with xo = 100. What is the value of z67?
Round your answer to two decimal places. Answer:
The value of [tex]z_{67}[/tex] is approximately 13.50 and by solving differential equation is [tex]X_{t+1} = 0.99,X_{t - 4}, X_0 = 100, X_1 = 95, X_2 = 90.05[/tex]
Given [tex]x_0 = 100[/tex] as the initial condition.
To solve the given difference equation:
[tex]X_{t+1} = 0.99 x_{t - 4}[/tex]
To find the values of [tex]X_t[/tex] recursively by substituting the previous term into the equation.
Calculate the values of [tex]X_t[/tex] for t = 0 to t = 67:
[tex]X_0 = 100[/tex] (given initial condition)
[tex]X_1 = 0.99 * X_0 - 4[/tex]
[tex]X_1 = 0.99 * 100 - 4[/tex]
[tex]X_1 = 99 - 4[/tex]
[tex]X_1 = 95[/tex]
[tex]X_2 = 0.99 * X_1 - 4[/tex]
[tex]X_2 = 0.99 * 95 - 4[/tex]
[tex]X_2 = 94.05 - 4[/tex]
[tex]X_2 = 90.05[/tex]
Continuing this process, and calculate [tex]X_t[/tex] for t = 3 to t = 67.
[tex]X_{67} = 0.99 * X_{66} - 4[/tex]
Using this recursive approach, find the value of [tex]X_{67}[/tex]. However, it is time-consuming to compute all the intermediate steps manually.
Alternatively, a formula to find the value of [tex]X_t[/tex] directly for any given t.
The general formula for the nth term of a geometric sequence with a common ratio of r and initial term [tex]X_0[/tex] is:
[tex]X_n = X_0 * r^n[/tex]
In our case, [tex]X_0 = 100[/tex] and r = 0.99.
Therefore, calculate [tex]X_{67}[/tex] as:
[tex]X_{67} = 100 * (0.99)^{67}[/tex]
[tex]X_{67} = 100 * 0.135[/tex]
[tex]X_{67} = 13.5[/tex]
Rounding to two decimal places,
[tex]X_{67}[/tex] ≈ 13.50
Therefore, the value of [tex]X_{67}[/tex] is approximately 13.50.
Therefore, the value of [tex]z_{67}[/tex] is approximately 13.50 and by solving differential equation is [tex]X_{t+1} = 0.99,x_{t - 4}, X_0 = 100, X_1 = 95, X_2 = 90.05[/tex]
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Let n = p1p2 .... pk where the pi are distinct primes. Show that µ(d) = (−1)^k µ (n/d)
The statement µ(d) = (−1)^k µ (n/d) relates to the Möbius function µ(d) and the prime factorization of an integer n. The Möbius function is a number-theoretic function that takes the value -1 if d is a square-free positive integer with an even number of prime factors, 0 if d is not square-free, and +1 if d is a square-free positive integer with an odd number of prime factors.
The prime factorization of n is given as n = p1p2....pk, where p1, p2, ..., pk are distinct prime numbers. The exponent of each prime pi in the factorization determines whether the number is square-free or not. If the exponent is even, the number is not square-free, and if the exponent is odd, the number is square-free.
The statement µ(d) = (−1)^k µ (n/d) can be proven by considering the cases where d is square-free and not square-free. If d is square-free, it means that the exponents of the prime factors in d are either 0 or 1. In this case, the Möbius function µ(d) will have the same value as µ(n/d), since the exponents cancel out.
On the other hand, if d is not square-free, it means that at least one of the exponents in d is greater than 1. In this case, both µ(d) and µ(n/d) will be equal to 0, as d is not a square-free positive integer.
Therefore, the statement µ(d) = (−1)^k µ (n/d) holds true, as it correctly reflects the relationship between the Möbius function and the prime factorization of an integer n. The exponent k in the equation represents the number of distinct prime factors in n.
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find a power series representation for the function and determine the interval of convergence. (give your power series representation centered at x = 0.)
f(x) = 1/6+x
Note that in this case,where the radius of convergence is 6, the interval of convergence is (-6, 6).
How is this so ?
To find the power series representation, we can use the following steps
Let f(x) = 1 /6+ x.
Let g(x) = f( x )- f(0).
Expand g(x) in a Taylor series centered at x = 0.
Add f(0) to the Taylor series for g(x).
The interval of convergence can be found using the ratio test. The ratio test says that the series converges if the limit of the absolute value of the ratio of successive terms is less than 1.
In this case, the limit of the absolute value of the ratio of successive terms is
lim_{n → ∞} |(x+6)/(n + 1)| = 1
Therefore, the interval of convergence is (-6, 6).
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Rewrite each of these statements in the form: V _____ x, ______
a. All Titanosaurus species are extinct. V_____ x,____ b. All irrational numbers are real.V_____ x,______ c. The number -7 is not equal to the square of any real number. V____ X, ____
Thus, we have rewritten each of the given statements in the form of V_____ x,_____.
The given statements are to be rewritten in the form: V_____ x,____.
a. All Titanosaurus species are extinct. V is “for all,” and x is “all Titanosaurus species.”
So, the statement is in the form of Vx. All Titanosaurus species are extinct can be written as:
Vx(Titanosaurus species are extinct).
b. All irrational numbers are real. V is “for all,” and x is “all irrational numbers.”
So, the statement is in the form of Vx. All irrational numbers are real can be written as:
Vx(Irrational numbers are real).
c. The number -7 is not equal to the square of any real number. V is “there exists,” and x is “any real number.”
So, the statement is in the form of Vx.
The number -7 is not equal to the square of any real number can be written as: ∃x(-7 ≠ x²).
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Question 3 (2 points) Test for differential patterns of church attendance (simple classification of whether each respondent has or has not attended a religious service within the past month) for 145 high school versus 133 college students, One Way Independent Groups ANOVA One Way Repeated Measures ANOVA Two Way Independent Groups ANOVA Two Way Repeated Measures ANOVA Two Way Mixed ANOVA Independent groups t-test
To test the differential patterns of church attendance for high school versus college students, we can use independent groups t-test. Here, we need to classify each respondent into two categories:
whether they have attended a religious service within the past month or not. In the t-test, we will compare the mean scores of church attendance for high school and college students and determine if the difference in means is statistically significant.
To conduct the independent groups t-test, we need to follow these steps:
Step 1: State the null and alternative hypotheses.H0: There is no significant difference in the mean scores of church attendance for high school and college students.H1: There is a significant difference in the mean scores of church attendance for high school and college students.
Step 2: Determine the level of significance.
Step 3: Collect data on church attendance for high school and college students.
Step 4: Calculate the means and standard deviations of church attendance for high school and college students.
Step 5: Compute the t-test statistic using the formula: [tex]t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^(1/2)[/tex], where x1 and x2 are the means of church attendance for high school and college students, s1 and s2 are the standard deviations of church attendance for high school and college students, and n1 and n2 are the sample sizes for high school and college students, respectively.
Step 6: Determine the degrees of freedom (df) using the formula: df = n1 + n2 - 2.
Step 7: Determine the critical values of t using a t-table or a statistical software program, based on the level of significance and degrees of freedom.
Step 8: Compare the calculated t-value with the critical values of t. If the calculated t-value is greater than the critical value, reject the null hypothesis. If the calculated t-value is less than the critical value, fail to reject the null hypothesis.
Step 9: Interpret the results and draw conclusions. In conclusion, we can use the independent groups t-test to test the differential patterns of church attendance for high school versus college students.
We need to classify each respondent into two categories: whether they have attended a religious service within the past month or not. The t-test compares the mean scores of church attendance for high school and college students and determines if the difference in means is statistically significant.
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Determine whether the lines below are parallel, perpendicular, or neither. - 6x – 2y = -10 y = 3x - 7 #15: Determine whether the lines below are parallel, perpendicular, or neither = y = 2x + 9 X – 2y = -6
The given lines are neither perpendicular nor parallel to each other. Hence, the correct option is option C.
The given equations of lines are -6x - 2y = -10 and y = 3x - 7.
To determine whether the given lines are parallel, perpendicular or neither; we need to convert both equations into a slope-intercept form that is y = mx + b, where m is the slope of the line and b is the y-intercept.
Therefore, y = 3x - 7 is already in slope-intercept form.
Let's convert -6x - 2y = -10 equation into slope-intercept form, which is:-2y = 6x - 10y = -3x + 5
So, the slope of the first line is -3 and the slope of the second line is 2.
As the slopes are different, the lines are not parallel to each other. Also, the product of the slope of both lines is -6 which is not equal to -1.
Therefore, the given lines are neither perpendicular nor parallel to each other. Hence, the correct option is option C.
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Evaluate the integral by making the given substitution.∫ dt /(1-6t)^4 u=1-6t
To evaluate the integral ∫ dt /[tex](1-6t)^{4}[/tex] using the given substitution u = 1-6t, we can rewrite the integral in terms of u. The resulting integral is ∫ (-1/6) du / [tex]u^{4}[/tex]. By simplifying and integrating this expression, we find the answer.
Let's start by making the given substitution u = 1-6t. To find the derivative of u with respect to t, we differentiate both sides of the equation, yielding du/dt = -6. Rearranging this equation, we have dt = -du/6.
Now, let's substitute these expressions into the original integral:
∫ dt /[tex](1-6t)^{4}[/tex] = ∫ (-du/6) /([tex]u^{4}[/tex]).
We can simplify this expression by factoring out the constant (-1/6):
(-1/6) ∫ du /[tex]u^{4}[/tex].
Now, we integrate the simplified expression. The integral of u^(-4) can be evaluated as [tex]u^{-3}[/tex] / -3, which gives us (-1/6) * (-1/3) * [tex]u^{-3}[/tex] + C.
Finally, we substitute the original variable u back into the result:
(-1/6) * (-1/3) * [tex](1-6t)^{-3}[/tex]+ C.
Therefore, the integral ∫ dt /[tex](1-6t)^{4}[/tex], evaluated using the given substitution u = 1-6t, is (-1/18) * [tex](1-6t)^{-3}[/tex]+ C.
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Simplify 4x* + 5x (x + 9) by factoring out x' 2 2 4x + 5x(x +9)= (Type your answer in factored form.) N/W
In order to simplify 4x² + 5x(x + 9) by factoring out x, first, you need to multiply 5x by the terms in the parentheses which is x + 9. This gives you 5x² + 45x. Then add 4x² to 5x² + 45x to obtain the simplified expression which is 9x² + 45x.
Step by step answer:
To simplify 4x² + 5x(x + 9) by factoring out x, follow the steps below;
Distribute the 5x in the parentheses to x and 9 in the following manner;
5x(x+9)=5x² + 45x
Add 4x² to 5x² + 45x which gives you;
4x² + 5x(x+9) = 4x² + 5x² + 45x
Simplify the above expression by adding like terms, 4x² and 5x²;4x² + 5x(x + 9) = 9x² + 45x
Factor out x from 9x² + 45x to obtain the final simplified expression which is; x(9x + 45) = 9x(x + 5)
Therefore, the simplified form of 4x² + 5x(x + 9) by factoring out x is 9x(x + 5).
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Answer a Question 1 [12] Evaluate the following 1.1 D2{xe*} 1.2 1 D²+2D+{cos3x} 1.3 // {x²} (D²²_4) { e²x} 2 [25] ing differen =
The evaluation of the given expressions is as follows:
1.1 D2{xe*} = 0
1.2 1 D²+2D+{cos3x} = -9cos(3x) - 6sin(3x) + cos(3x)
1.3 // {x²} (D²²_4) { e²x} = 0
First, let's find the first derivative of xe*. Using the product rule, the derivative of xe* is given by (1e) + (x * d/dx(e*)), where d/dx denotes the derivative with respect to x. Since d/dx(e*) is simply 0 (the derivative of a constant), the first derivative simplifies to e*.
Now, let's find the second derivative of xe*. Applying the product rule again, we have (1 * d/dx(e*)) + (x * d²/dx²(e*)). As mentioned earlier, d/dx(e*) is 0, so the second derivative simplifies to 0.
Therefore, the evaluation of D2{xe*} is 0.
1.2 1 D²+2D+{cos3x}:
The expression 1 D²+2D+{cos3x} represents the differential operator acting on the function 1 + cos(3x). To evaluate this expression, we need to apply the given differential operator to the function.
The differential operator D²+2D represents the second derivative with respect to x plus two times the first derivative with respect to x.
First, let's find the first derivative of 1 + cos(3x). The derivative of 1 is 0, and the derivative of cos(3x) with respect to x is -3sin(3x). Therefore, the first derivative of the function is -3sin(3x).
Next, let's find the second derivative. Taking the derivative of -3sin(3x) with respect to x gives us -9cos(3x). Hence, the second derivative of the function is -9cos(3x).
Now, we can evaluate the expression 1 D²+2D+{cos3x} by substituting the second derivative (-9cos(3x)) and the first derivative (-3sin(3x)) into the expression. This gives us 1 * (-9cos(3x)) + 2 * (-3sin(3x)) + cos(3x), which simplifies to -9cos(3x) - 6sin(3x) + cos(3x).
Therefore, the evaluation of 1 D²+2D+{cos3x} is -9cos(3x) - 6sin(3x) + cos(3x).
1.3 // {x²} (D²²_4) { e²x}:
The expression // {x²} (D²²_4) { e²x} represents the composition of the differential operator (D²²_4) with the function e^(2x) divided by x².
First, let's evaluate the differential operator (D²²_4). The notation D²² represents the 22nd derivative, and the subscript 4 indicates that we need to subtract the fourth derivative. However, since the function e^(2x) does not involve any x-dependent terms that would change upon differentiation, the derivatives will all be the same. Therefore, the 22nd derivative minus the fourth derivative of e^(2x) is simply 0.
Next, let's divide the result by x². Dividing 0 by x² gives us 0.
Therefore, the evaluation of // {x²} (D²²_4) { e²x} is 0.
In summary, the evaluation of the given expressions is as follows:
1.1 D2{xe*} = 0
1.2 1 D²+2D+{cos3x} = -9cos(3x) - 6sin(3x) + cos(3x)
1.3 // {x²} (D²²_4) { e²x} = 0
The first expression represents the second derivative of xe*, which simplifies to 0. The second expression involves applying a given differential operator to the function 1 + cos(3x), resulting in -9cos(3x) - 6sin(3x) + cos(3x). The third expression represents the composition of a differential operator with the function e^(2x), divided by x², and simplifies to 0.
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Consider the following IVP: x' (t) = -x (t), x (0)=xo¹ where λ= 23 and x ER. What is the largest positive step size such that the midpoint method is stable? Write your answer to three decimal places. Hint: Follow the same steps that we used to show the stability of Euler's method. Step 1: By iteratively applying the midpoint method, show y₁ =p (h) "xo' where Step 2: Find the values of h such that lp (h) | < 1. p(h) is a quadratic polynomial in the step size, h. Alternatively, you can you could take a bisection type approach where you program Matlab to use the midpoint method to solve the IVP for different step sizes. Then iteratively find the largest step size for which the midpoint method converges to 0 (be careful with this approach because we are looking for 3 decimal place accuracy).
So the largest positive step size such that the midpoint method is stable is 1.
We are supposed to consider the following IVP: x' (t) = -x (t), x (0)=xo¹ where λ= 23 and x ER.
We are to find the largest positive step size such that the midpoint method is stable.
Step 1: By iteratively applying the midpoint method, show y₁ =p (h) "xo' where
Using midpoint method
y1=yo+h/2*f(xo, yo)y1=xo+(h/2)*(-xo)y1=xo*(1-h/2)
Therefore,y1=p(h)*xo where p(h)=1-h/2Thus,y1=p(h)*xo ......(1)
Step 2: Find the values of h such that lp (h) | < 1.
p(h) is a quadratic polynomial in the step size, h.
From equation (1), we have
y1=p(h)*xo
Let y0=1
Then y1=p(h)*y0
The characteristic equation is given by
y₁ = p(h) y₀y₁/y₀ = p(h)Hence λ = p(h)
So,λ=1-h/2Now,lp(h)l=|1-h/2|
Assuming lp(h)<1=⇒|1-h/2|<1
We need to find the largest positive step size such that the midpoint method is stable.
For that we put |1-h/2|=1h=1
Hence, the required solution is 1.
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1% of the electric bulbs that is produced by a factory are defective. In a random sample of 250 electric bulbs, find the probability that 3 electric bulbs are defective.
To find the probability that exactly 3 electric bulbs are defective, we can use the binomial probability formula.
The probability of success (defective bulb) is 1% or 0.01, and the probability of failure (non-defective bulb) is 99% or 0.99. Plugging in these values into the formula, we have P(X = 3) = (250 choose 3) * 0.01^3 * 0.99^(250-3), where (250 choose 3) represents the combination of choosing 3 bulbs out of 250. Evaluating this expression gives us the desired probability. The probability that exactly 3 electric bulbs are defective in a random sample of 250 bulbs can be calculated using the binomial probability formula. By plugging in the values for the probability of success (defective bulb) and failure (non-defective bulb), along with the combination of choosing 3 bulbs out of 250, we can determine the probability.
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Lett be the 7th digit of your Student ID. Consider the utility function u(r, g) = 1 t+2 -In(1+x) + 1 t+2 zln(1 + y) (a) [10 MARKS] Compute the Hessian matrix D²u(x, y). Is u concave or convex? (b) [4 MARKS] Give the formal definition of a convex set. (c) [8 MARKS] Using your conclusion to (a), show that I+(1) = {(x, y) = R²: u(x, y) ≥ 1} is a convex set. (d) [8 MARKS] Compute the 2nd order Taylor polynomial of u(x, y) at (0,0).
A Hessian matrix, D²u(x, y), is a square matrix consisting of second-order partial derivatives of a multivariable function. The matrix is symmetric by definition, so it suffices to compute half of the matrix. To verify whether the function u(r, g) is convex or concave, we'll use the Hessian matrix's determinants.
Thus, we can conclude that the Hessian matrix of the function u(r, g) is positive semi-definite. Hence, the function is a concave function.(a) We will take the second derivative of u with respect to each variable to compute the Hessian matrix. Here are the second derivatives of u:$$\begin{aligned} \frac{\partial u}{\partial x^2} &= \frac{2}{(1+x)^2} &\qquad \frac{\partial^2 u}{\partial x\partial y} &= 0 \\ \frac{\partial^2 u}{\partial y\partial x} &= 0 &\qquad \frac{\partial u}{\partial y^2} &= \frac{2z}{(1+y)^2} \end{aligned}$$Thus, the Hessian matrix D²u(x, y) is:$$D^2u(x, y)=\begin{pmatrix} \frac{2}{(1+x)^2} & 0 \\ 0 & \frac{2z}{(1+y)^2} \end{pmatrix}$$Since both diagonal entries of the matrix are positive, the function u(r, g) is concave.(b) A convex set is defined as follows:A set C in Rn is said to be convex if for every x, y ∈ C and for all t ∈ [0, 1], tx + (1 − t)y ∈ C.It means that all points on a line segment connecting two points in the set C should also be in C. That is, any line segment between any two points in C should be contained entirely in C.(c)We will use the Hessian matrix's positive semi-definiteness to show that I+(1) = {(x, y) = R²: u(x, y) ≥ 1} is a convex set.If D²u(x, y) is positive semi-definite, it means that the eigenvalues are greater than or equal to zero. The eigenvalues of D²u(x, y) are:$$\lambda_1 = \frac{2}{(1+x)^2} \quad \text{and} \quad \lambda_2 = \frac{2z}{(1+y)^2}$$Since both eigenvalues are greater than or equal to zero, D²u(x, y) is positive semi-definite. As a result, the set I+(1) is convex because u(x, y) is a concave function.(d) The second-order Taylor polynomial of u(x, y) at (0, 0) is given by:$$u(0,0)+\begin{pmatrix} 0 \\ 0 \end{pmatrix}^T \nabla u(0,0)+\frac{1}{2}\begin{pmatrix} 0 \\ 0 \end{pmatrix}^T D^2u(0,0)\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$$$=u(0,0)+0+0=1$$Therefore, the 2nd order Taylor polynomial of u(x, y) at (0,0) is 1.
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A Hessian matrix, [tex]D^{2} u(x, y)[/tex], is a square matrix consisting of second-order partial derivatives of a multivariable function. The matrix is symmetric by definition, so it suffices to compute half of the matrix. To verify whether the function u(r, g) is convex or concave, we'll use the Hessian matrix's determinants.
Here, we have,
Thus, we can conclude that the Hessian matrix of the function u(r, g) is positive semi-definite. Hence, the function is a concave function.
(a) We will take the second derivative of u with respect to each variable to compute the Hessian matrix.
Here are the second derivatives of u:
{∂ u}/{∂ x²} = {2}/{(1+x)²}
{∂² u}/{∂ x∂ y} = 0
{∂² u}/{∂ y∂ x} = 0
{∂ u}/{∂ y²} = {2z}/{(1+y)²}
Thus, the Hessian matrix [tex]D^{2} u(x, y)[/tex] is:
[tex]D^{2} u(x, y)[/tex]=[tex]\begin{pmatrix} \frac{2}{(1+x)²} & 0 \\ 0 & \frac{2z}{(1+y)²} \end{pmatrix}[/tex]
Since both diagonal entries of the matrix are positive, the function u(r, g) is concave.
(b) A convex set is defined as follows:
A set C in Rn is said to be convex if for every x, y ∈ C and for all t ∈ [0, 1], tx + (1 − t)y ∈ C.
It means that all points on a line segment connecting two points in the set C should also be in C.
That is, any line segment between any two points in C should be contained entirely in C.
(c)We will use the Hessian matrix's positive semi-definiteness to show that I+(1) = {(x, y) = [tex]R^{2}[/tex]: [tex]u(x, y)\geq 1[/tex]} is a convex set.
If [tex]D^{2} u(x, y)[/tex] is positive semi-definite, it means that the eigenvalues are greater than or equal to zero.
The eigenvalues of [tex]D^{2} u(x, y)[/tex] are:
[tex]\lambda_1 = \frac{2}{(1+x)²} \quad \text{and} \quad \lambda_2 = \frac{2z}{(1+y)²}[/tex]
Since both eigenvalues are greater than or equal to zero,[tex]D^{2} u(x, y)[/tex] is positive semi-definite. As a result, the set I+(1) is convex because u(x, y) is a concave function.
(d) The second-order Taylor polynomial of u(x, y) at (0, 0) is given by:
[tex]u(0,0)+\begin{pmatrix} 0 \\ 0 \end{pmatrix}^T \nabla u(0,0)+\frac{1}{2}\begin{pmatrix} 0 \\ 0 \end{pmatrix}^T D²u(0,0)\begin{pmatrix} 0 \\ 0 \end{pmatrix}=u(0,0)+0+0=1[/tex]
Therefore, the 2nd order Taylor polynomial of u(x, y) at (0,0) is 1.
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4. Let's assume the ages at retirement for NFL football players is normally distributed, with μ = 35 and o = 2 years of age.
(a) How likely is it that a player retires after their 40th birthday?
(b) What is the probability a player retires before the age of 26?
(c) What is the probability a player retires between ages o30 and 35?
(a) The likeliness of a player to retire after their 40th birthday is approximately 0.0062 or 0.62%.
(b) The probability that a player retires before the age of 26 is approximately zero..
(c) The probability that a player retires between ages 30 and 35 is approximately 0.4938 or 49.38%.
(a) The given normal distribution has a mean (μ) of 35 and standard deviation (σ) of 2. We need to find the probability that a player retires after their 40th birthday.
z = (x - μ)/σ, where x = 40. z = (40 - 35)/2 = 2.5
Using the standard normal distribution table, we can find the probability that a z-score is less than 2.5 (because we need the probability of a player retiring after their 40th birthday). The table gives a probability of 0.9938.
So, the probability that a player retires after their 40th birthday is approximately 0.0062 or 0.62%.
(b) Here, we need to find the probability that a player retires before the age of 26. Again, using the standard normal distribution, z = (x - μ)/σ, where x = 26. z = (26 - 35)/2 = -4.5
We need to find the probability that a z-score is less than -4.5 (because we need the probability of a player retiring before the age of 26). This is a very small probability, which we can estimate as zero.
So, the probability that a player retires before the age of 26 is approximately zero.
(c) In this case, we need to find the probability that a player retires between ages 30 and 35. We can use the standard normal distribution again.
z1 = (30 - 35)/2 = -2.5
z2 = (35 - 35)/2 = 0
The probability that a z-score is between -2.5 and 0 can be found using the standard normal distribution table. This probability is approximately 0.4938.
So, the probability that a player retires between ages 30 and 35 is approximately 0.4938 or 49.38%.
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What is the value of x?
sin x° = cos 50°
ОС
100
50
40
130
90
The value of x is 40°.
To find the value of x, we need to determine the angle whose sine is equal to the cosine of 50°.
Since the sine of an angle is equal to the cosine of its complementary angle, we can use the complementary angle relationship to solve the equation.
The complementary angle of 50° is 90° - 50° = 40°.
Therefore, the value of x is 40°.
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Evaluate the indefinite integral. Use a capital "C" for any constant term
∫( 4e^x – 2x^5+ 3/x^5-2) dx )
we add up all the integrals and the respective constant terms to obtain the complete solution: 4e^x + (1/3)x^6 - 3/(4x^4) + 2x + C.∫(4e^x – 2x^5 + 3/x^5 - 2) dx.
To evaluate the indefinite integral of the given expression, we will integrate each term separately.
∫4e^x dx = 4∫e^x dx = 4e^x + C1
∫2x^5 dx = 2∫x^5 dx = (2/6)x^6 + C2 = (1/3)x^6 + C2
∫3/x^5 dx = 3∫x^-5 dx = 3(-1/4)x^-4 + C3 = -3/(4x^4) + C3
∫2 dx = 2x + C4
Putting all the terms together, we have:
∫(4e^x – 2x^5 + 3/x^5 - 2) dx = 4e^x + (1/3)x^6 - 3/(4x^4) + 2x + C
where C = C1 + C2 + C3 + C4 is the constant of integration.
In the given problem, we are asked to find the indefinite integral of the expression 4e^x – 2x^5 + 3/x^5 - 2 dx.
To solve this, we integrate each term separately and add the resulting integrals together, with each term accompanied by its respective constant of integration.
The first term, 4e^x, is a straightforward integral. We use the rule for integrating exponential functions, which states that the integral of e^x is e^x itself. So, the integral of 4e^x is 4 times e^x.
The second term, -2x^5, involves a power function. Using the power rule for integration, we increase the exponent by 1 and divide by the new exponent. So, the integral of -2x^5 is (-2/6)x^6, which simplifies to (-1/3)x^6.
The third term, 3/x^5, can be rewritten as 3x^-5. Applying the power rule, we increase the exponent by 1 and divide by the new exponent. The integral of 3/x^5 is then (-3/4)x^-4, which can also be written as -3/(4x^4).
The fourth term, -2, is a constant, and its integral is simply the product of the constant and x, which gives us 2x.
Finally, we add up all the integrals and the respective constant terms to obtain the complete solution: 4e^x + (1/3)x^6 - 3/(4x^4) + 2x + C. Here, C represents the sum of the constant terms from each integral and accounts for any arbitrary constant of integration.
Note: In the solution, the constants of integration are denoted as C1, C2, C3, and C4 for clarity, but they are ultimately combined into a single constant, C.
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A survey was taken asking the favorite flavor of coffee drink a person prefers. The responses were: V = vanilla, C= caramel, M= mocha, H-hazelnut, P=plain. Construct a categorical frequency distribution for the data. Which class has the most data and which has the least. Also construct a pie chart and a cumulative frequency chart for this data.
Data for 5:
V C P P M M P P M C
M M V M M M V M M M
P V C M V M C P M P
M M M P M M C V M C
C P M P M H H P H P
To construct a categorical frequency distribution for the given data, we will count the number of occurrences for each flavor category. Here's the frequency distribution:
From the frequency distribution, we can determine that the flavor category "M" has the most data with a frequency of 14. On the other hand, the flavor category "H" has the least data with a frequency of 3 In the pie chart, each flavor category is represented by a sector, and the size of each sector corresponds to the frequency of that flavor category. The largest sector represents the flavor "M," which is the most preferred coffee flavor. The smallest sector represents the flavor "H," which is the least preferred coffee flavor , the cumulative frequency chart, the cumulative frequency for each flavor category is calculated by adding up the frequencies from the beginning of the distribution to that particular category. It provides a visual representation of the cumulative data as we move through the flavors
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Let {X(t), t = [0, [infinity]0)} be defined as X(t) = A + Bt, for all t = [0, [infinity]), where A and B are independent normal N(1, 1) random variables. a. Find all possible sample functions for this random proces.
b. Define the random variable Y = X(1). Find the PDF of Y. c. Let also Z = X(2). Find E[YZ].
The random process X(t) = A + Bt, where A and B are independent normal random variables with mean 1 and variance 1, has an infinite set of possible sample functions.
a. The sample functions of the random process X(t) = A + Bt are obtained by substituting different values of t into the expression. Since A and B are independent normal random variables, each sample function is a linear function of t with coefficients A and B. Therefore, the set of possible sample functions is infinite.
b. To find the PDF of the random variable Y = X(1), we substitute t = 1 into the expression for X(t). We get Y = A + B, which is a linear combination of two independent normal random variables. The sum of normal random variables is also normally distributed, so Y follows a normal distribution. The mean of Y is the sum of the means of A and B, which is 1 + 1 = 2. The variance of Y is the sum of the variances of A and B, which is 1 + 1 = 2. Hence, the PDF of Y is a normal distribution with mean 2 and variance 2.
c. The expected value of the product of Y and Z, denoted as E[YZ], can be calculated as E[YZ] = E[X(1)X(2)]. Since X(t) = A + Bt, we have X(1) = A + B and X(2) = A + 2B. Substituting these values, we get E[YZ] = E[(A + B)(A + 2B)]. Expanding and simplifying, we find E[YZ] = E[[tex]A^2[/tex] + 3AB + 2[tex]B^2[/tex]]. Since A and B are independent, their cross-product term E[AB] is zero. The expected values of [tex]A^2[/tex] and [tex]B^2[/tex] are equal to their variances, which are both 1. Thus, E[YZ] simplifies to E[[tex]A^2[/tex]] + 3E[AB] + 2E[[tex]B^2[/tex]] = 1 + 0 + 2 = 3. Therefore, the expected value of YZ is 3.
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A random sample of 1,000 peope was taken. Six hundred fifty of the people in the sample favored candidate A. What is the 95% confidence interval for the true proportion of people who favor Candidate A?
a) 0.600 to 0.700
b) 0.620 to 0.680
c) 0.623 to 0.678
d) 0.625 to 0.675
At a 95% confidence interval, 0.623–0.678 proportion of people favor Candidate A.
A random sample of 1,000 people was taken. Six hundred fifty of the people in the sample favored candidate A. Confidence interval = point estimate ± margin of error. Here, the point estimate is the sample proportion. It is given by: Point estimate = (number of people favoring candidate A) / (total number of people in the sample)= 650/1000= 0.65. The margin of error is given by: Margin of error = z* sqrt(p(1-p)/n). Here, p is the proportion of people favoring candidate A and n is the sample size, and z* is the z-score corresponding to the 95% confidence level. The value of z* can be obtained using a z-table or a calculator. Here, we will assume it to be 1.96 since the sample size is large, n > 30. So, the margin of error is given by: Margin of error = 1.96 * sqrt(0.65 * 0.35 / 1000)≈ 0.028. So, the 95% confidence interval for the true proportion of people who favor Candidate A is given by: 0.65 ± 0.028= (0.622, 0.678)Therefore, the correct option is c) 0.623 to 0.678.
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21. DETAILS LARPCALC10CR 1.4.030. Find the function value, if possible. (If an answer is undefined, enter UNDEFINED.) x < -1 -4x-4, x²+2x-1, x2-1 (a) f(-3) (b) (-1) (c) f(1) DETAILS LARPCALC10CR 3.4.
The function values for the given equation are as follows:
(a) f(-3) = -4
(b) f(-1) = -4
(c) f(1) = 4
What are the function values for x = -3, -1, and 1?The function values for the given equation can be calculated as follows:
(a) f(-3): Substitute x = -3 into the equation -4x-4:
f(-3) = -4(-3) - 4
= 12 - 4
= 8
(b) f(-1): Substitute x = -1 into the equation x²+2x-1:
f(-1) = (-1)² + 2(-1) - 1
= 1 - 2 - 1
= -2
(c) f(1): Substitute x = 1 into the equation x²-1:
f(1) = 1² - 1
= 1 - 1
= 0
Therefore, the function values are:
(a) f(-3) = 8
(b) f(-1) = -2
(c) f(1) = 0
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Select your answer What is the center of the shape formed by the equation (x-3)² (y+5)² 49 = 1? 25 ○ (0,0) O (-3,5) O (3,-5) O (9,25) (9 out of 20) (-9, -25)
The answer is , the correct option is \[\boxed{\mathbf{(C)}\ (3,-5)}\].
How to find?The equation of the ellipse can be rewritten in standard form as:
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]
where (h, k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.
The equation \[(x-3)^2(y+5)^2/49 = 1\] represents an ellipse with center at \[(3,-5)\].
Since the center of the ellipse formed by the equation \[(x-3)^2(y+5)^2/49 = 1\] is \[(3,-5)\], the answer is \[(3,-5)\].
Hence, the correct option is \[\boxed{\mathbf{(C)}\ (3,-5)}\].
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The perimeter of a rectangle is equal to the sum of the lengths of the four sides. If the length of the rectangle is L and the width of the rectangle is W, the perimeter can be written as: 2L + 2W Suppose the length of a rectangle is L = 6 and its width is W = 5. Substitute these values to find the perimeter of the rectangle.
The perimeter of the rectangle is 22 units supposing the length of a rectangle is L = 6 and its width is W = 5.
A rectangle's perimeter is determined by adding the lengths of its four sides. The perimeter of a rectangle of length L and width W can be expressed mathematically as 2L + 2W. Let's say a rectangle has a length of 6 and a width of 5. Substituting these values into the formula for the perimeter of the rectangle, we have: Perimeter = 2L + 2W= 2(6) + 2(5)= 12 + 10= 22 units. Therefore, the perimeter of the rectangle is 22 units.
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Set up the objective function and the constraints, but do not solve.
Home Furnishings has contracted to make at least 295 sofas per week, which are to be shipped to two distributors, A and B. Distributor A has a maximum capacity of 140 sofas, and distributor B has a maximum capacity of 160 sofas. It costs $14 to ship a sofa to A and 512 to ship to B. How many sofas should be produced and shipped to each distributor to minimize shipping costs? (Let x represent the number of sofas shipped to Distributor A, y the number of sofas shipped to Distributor B, and z the shipping costs in dollars.) -
Select- = subject to
required sofas ___
distributor A limitation ___
distributor B limitation ___
x > 0, y > 0
The subject to required sofas ≥ 295x ≤ 140y ≤ 160x > 0, y > 0
Distributor A limitation x ≤ 140
Distributor B limitation y ≤ 160x > 0, y > 0
Objective Function and ConstraintsA Home Furnishing company is contracted to make 295 or more sofas per week. These sofas are to be shipped to two distributors, A and B. In order to minimize the shipping costs, the company is tasked with finding the optimal number of sofas to ship to each distributor.
Let x represent the number of sofas shipped to Distributor A, y the number of sofas shipped to Distributor B, and z the shipping costs in dollars.The objective function:
Minimize Z = 14x + 12y (Since it costs $14 to ship a sofa to A and $12 to ship to B)
Subject to: required sofas ≥ 295
distributor A limitation: x ≤ 140
distributor B limitation: y ≤ 160x > 0, y > 0 (As negative numbers of sofas are not possible)
Therefore, the objective function and constraints are:
Minimize Z = 14x + 12y
Subject to:required sofas ≥ 295x ≤ 140y ≤ 160x > 0, y > 0
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MAT123 Spring 2022 HW 6, Due by May 30 (Monday), 10:00 PM (KST) log4(x + 2) + log, 3 = log4 5+ log.(2x - 3) Problem 3 [Logarithmic Equations] Solve the logarithmic equation algebraically.
The simplified logarithmic equation is x = 1/2.
To solve the given logarithmic equation algebraically, we need to eliminate the logarithms by applying logarithmic properties. Let's break down the solution into three steps.
Use the logarithmic properties to combine the logarithms on both sides of the equation. Applying the product rule of logarithms, we get:
log4(x + 2) + log3 = log4(5) + log(2x - 3)
Apply the power rule of logarithms to simplify further. According to the power rule, logb(a) + logb(c) = logb(ac). Using this rule, we can rewrite the equation as:
log4[(x + 2) * 3] = log4(5 * (2x - 3))
Simplifying both sides:
log4(3x + 6) = log4(10x - 15)
Step 3:
Now that the logarithms have been eliminated, we can equate the expressions within the logarithms. This gives us:
3x + 6 = 10x - 15
Solving for x, we can simplify the equation:
7x = 21
x = 3
Therefore, the main answer to the given logarithmic equation is x = 3/7.
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2. find the component of a in the direction of b, find the projection of a in the direction of b.
a = [1, 1, 1]; b = [2, 0, 1]
The component of a in the direction of b is approximately [0.8, 0, 0.4] and the projection of a onto b is [1.6, 0, 0.8]
To calculate the component of vector a in the direction of vector b, we need to find the projection of vector a onto vector b. The projection of a onto b represents the shadow of a cast in the direction of b. Mathematically, the projection of a onto b can be calculated as follows:
projection of a onto b = (dot product of a and b) / (magnitude of b)
In this case, the dot product of a = [1, 1, 1] and b = [2, 0, 1] is:
a · b = 1 * 2 + 1 * 0 + 1 * 1 = 3
The magnitude of b can be found using the formula:
magnitude of b = √(2^2 + 0^2 + 1^2) = √5
Therefore, the projection of a onto b is:
projection of a onto b = 3 / √5 ≈ [1.6, 0, 0.8]
This projection represents the component of a in the direction of b. The x-component of the projection is 1.6, the y-component is 0, and the z-component is 0.8. Hence, the component of a in the direction of b is approximately [0.8, 0, 0.4].
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The slope field for the equation y = -x +y is shown below 11:11 1-1-1-1 TTTTTTIT 1 - - 1 - 1 - 3 - 4 - 3- 4-4-4-4- 1411111 1111 On a print out of this slope field, sketch the solutions that pass through the points (i) (0,0); (ii) (-3,1); and (iii) (-1,0). From your sketch, what is the equation of the solution to the differential equation that passes through (-1,0)? (Verify that your solution is correct by substituting it into the differential equation.) y = }}}}}} ///// }}}}}/ 7171/ }}}} 3.12. Match each differential equation to a function which is a solution. FUNCTIONS A. y = 3x + x², B. y = e-8, C. y = sin(x), D.y=xt, E. y = 3 exp(2x), DIFFERENTIAL EQUATIONS 1. xy - y = x² 2. y"+y=0 3. y" + 15y +56y = 0 4.2x²y" + 3xy = y
The matched differential equations with their corresponding functions are:
xy - y = x² → y = x² (C)y" + y = 0 → y = Acos(x) + Bsin(x) (where A and B are constants)(C)y" + 15y + 56y = 0 → y = [tex]Ae^(-7x) + Be^(-8x)[/tex](where A and B are constants)(B)2x²y" + 3xy = y → y = [tex]Ax^(-1) + Bx^(-2)[/tex] (where A and B are constants)(D)Given that the slope field for the equation y = -x + y is shown below and we have to sketch the solutions that pass through the points (i) (0,0); (ii) (-3,1); and (iii) (-1,0).
From the sketch, we need to find the equation of the solution to the differential equation that passes through (-1,0).The slope field for the equation y = -x + y is shown below:
As shown in the slope field, the slope of the differential equation y = -x + y can be given as:dy/dx = y - x
The solution that passes through the point (0, 0) is y = x.
The solution that passes through the point (-3, 1) is y = x - 1.
The solution that passes through the point (-1, 0) is y = x.
The equation of the solution to the differential equation that passes through (-1, 0) is y = x.
To verify that our solution is correct, we need to substitute y = x in the differential equation:
dy/dx = y - x
dy/dx = x - x
dy/dx = 0
Therefore, y = x is a solution of the differential equation.
The differential equation that matches with the given functions are:1. xy - y = x² will have a function y = x²(C)
2. y" + y = 0 will have a function y = Acos(x) + Bsin(x)(where A and B are constants)(C)
3. y" + 15y + 56y = 0 will have a function [tex]y = Ae^(-7x) + Be^(-8x)[/tex](where A and B are constants)(B)
4. 2x²y" + 3xy = y will have a function[tex]y = Ax^(-1) + Bx^(-2)[/tex](where A and B are constants)(D)
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