Please solve the following questions
(1) (a) Find the order, Degree and linear or Non-Linear Differential Equations for d 2 = V2x + y. (1 Mark) (b) verify the functions sin(3x) and cos(3x) are linear dependent or Linear independent. (1 Mark) (2) Construct a Differential Equation for the given equation y = a + bsinx., by eliminating the arbitrary constants a and b. (2 Mark) (3) Solve the given differential equation (x* + y*)dx = xy'dy. (2Mark) (4) Solve the initial value problem " = 2x2 - 2, dx y(1) = 1. (2Mark) (5) Find the general solution for the differential equation (x2 - 4xy - 2y?) dx + (y2 - 4xy - 2x2 ) dy. (2 Mark)In class I showed that if the probability of one person to vote for Candidate A was 0.4 (and the probability they do not vote for A, but for candidate B instead is 0.6), then the probability that exactly x out of 3 people would vote for candidate A is given by the binomial probability formulas in the following table: X Pr(exactly x out of 3) 3! 0! (3 - 0)! -0.40.6(3-0) 1 3! -0.4 0.6(3-1) 1! (3 - 1)! 3! 2! (3 - 2)! -0.420.6(3-2) 3 3! 3! (3 - 3)! 0.430.6(3-3) For this extra credit assignment, you are to 1) Calculate the probabilities for each X and fill in the blank column on the right above. (Remember that c = 1, where c is any constant number not equal to either 0 or tom. Also remember that x! = x(x-1)(x-2)..3(2)(1).) You can use either an advanced calculator or StatCrunch for these probabilities. 2) Use the E(X) grouped mean formula from Ch. 6.1: ),=, x; p(x;), to find the expected value of the binomial variable X above. Show your work below. 3) Use the shortcut formula for E(X) of a binomial (E(X) = np) and show that the two different ways of calculating E(X) are identical FOR THIS PARTICULAR BINOMIAL RANDOM VARIABLE. Show your work below 4) Use the Var(X) grouped variance formula from Ch. 6.1: ), (x, - mp)-p(x;), to find the expected value of the binomial variable X above. Show your work below. 5) Use the shortcut formula for Var(X) of a binomial (Var(X) = np(1-p)) and show that the two different ways of calculating Var(X) are identical FOR THIS PARTICULAR BINOMIAL RANDOM VARIABLE. Show your work below. 6) Similarly, use the Var(X) grouped variance formula from Ch. 6.1: )=, (x; - # ) p(x,), to find the variance of the binomial random variable in the above table. (Remember that E(X) can also be written as u when using this formula, because E(X) is a grouped mean. Also note that you can use the Mean Squared Deviation way of looking at the Variance, but now the Mean is given by multiplying by P(x) rather than 1 as in Ch. 2, ] Show your work below 7) Use the shortcut formula for Var(X) of a binomial (Var(X) = np(1-p)) to find the variance of the binomial random variable X above. Show your work below. Your answers to 2 and 3 should be equal, and your answers to 4 and 5 should be equal. If they are not equal, either you have made a mistake in calculations using the Ch. 6.1 formulas, or you have rounded too much for each calculation. Please correct your errors so that the answers are equal - and correct!3. Show whether the good is normal or inferior good, and whether ordinary or Giffen good when its demand function is given as follows. (Hint: Think of the formal condition related to each category.) (each 2 points, total 6 points) 1) *= m - 3p 2 x=2+-D 2) 5 3) *=2p+4m
