y"+3xy'-4y=0

Solve the differential equation.

Differential Equations

Given that ?? = ?? is a solution of the given ODE (a) use reduction of order to find the general solution of the ODE ????? − 5??? + 9? = 0; (b) what is the second linearly independent solution, ??, of the ODE?

Consider only square matrices. True or false? Explain.

a) det(AB) = (det A)(det B).

b) Eigenvalues of all real matrices are real.

c) The determinant of an upper triangular matrix is the sum of its main diagonal entries.

Given z = 5 - 2i. Determine the following: a) Re(z) b) Im(z) c) Arg(z) d) arg(z) e) conjugate(z) f) modulus(z) g) z in polar form h) z in exponential form i) z^2 in polar form.

Solve by using power series: 2 y'−y = sinh( x). Find the recurrence relation and compute the first 6 coefficients (a0-a5). Use the methods of chapter 3 to solve the differential equation and show your chapter 8 solution is equivalent to your chapter 3 solution.

Belmont and Marolla conducted a study on the relationship between birth order, family size, and intelligence. The subjects consisted of all Dutch men who reached the age of 19 between 1963 and 1966. These men were required by law to take the Dutch army induction tests, including Raven’s intelligence test. The results showed that for each family size, measured intelligence decreased with birth order: first-borns did better than second-borns, second-borns did better than third-borns, and so on. And for any particular birth order, intelligence decreased with family size: for instance, first-borns in two-child families did better than firstborns in three-child families. Taking, for instance, men from two-child families: • the first-borns averaged 2.58 on the test; • the second-borns averaged 2.68 on the test. (Raven test scores range from 1 to 6, with 1 being best and 6 worst.) The difference is small, but if it is real, it has interesting implications for genetic theory. To show that the difference was real, Belmont and Marolla made a twosample t-test. The standard deviation for the test scores was around one point in both groups, and there were 30,000 men in each group. Belmont and Marolla concluded: “Thus the observed difference was highly significant . . .a high level of statistical confidence can be placed in each average because of the large number of cases.” Do you agree with their conclusion? Why or why not? Was it appropriate to make a two-sample t-test in this situation? (15 points)

List 3 different ways of calculating the Jacobian for a manipulator arm. Describe in words what happens to the degrees of freedom of a manipulator at configurations where Jacobian becomes singular.

Consider the 90 degrees rotation matrix R = [0 −1 1 0]

a) Are the eigenvalues real?

b) Are the eigenvectors real?

c) Find the determinant of R.

d) Find the trace of R.

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x²y²+z²).

a) Find rate of change of T at the point P(1,-1,-1) in the direction of Q(2,0,0)?

b) In which direction from P(1,-1,-1) does the temperature T increase most rapidly?

c) What is the maximum rate of change of T at P(1,-1,-1)?

Let (V, ||·||) be a normed space, and W a d^Norm_V,||·|| -closed vector subspace of V.

(a) Prove that a function |||·||| : V /W → R_≥0 can be consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| : R_≥0 | w ∈ W}).

(b) Prove that |||·||| is a norm on V /W.

(c) Prove that if (V, ||·||) is a Banach space, then so is (V /W, |||·|||)

A spherical snowball is melting in such a way that its radius is decreasing at rate of 0.1 cm/min. At what rate is the volume of the snowball decreasing when the radius is 14 cm. (Note the answer is a positive number).

When air expands adiabatically (without gaining or losing heat), its pressure PP and volume VV are related by the equation PV1.4=CPV1.4=C where CC is a constant. Suppose that at a certain instant the volume is 310310 cubic centimeters and the pressure is 8181 kPa and is decreasing at a rate of 1414 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?

A company's revenue from selling x units of an item is given as R=1700x−1x2R=1700x-1x2. If sales are increasing at the rate of 20 units per day, how rapidly is revenue increasing (in dollars per day) when 150 units have been sold?

A company selling widgets has found that the number of items sold, x, depends upon the price, p at which they're sold, according the equation x=20000√2p+1x=200002p+1

Due to inflation and increasing health benefit costs, the company has been increasing the price by $2 per month. Find the rate at which revenue is changing when the company is selling widgets at $160 each.

We look at the inner product space of continuous functions above [0,1].
a) Calculate the angle between x and cosx ; x and sinx. Which angle is the smallest?
b) Calculate the distance between x and cosx ; x and sinx. What distance is the shortest? Hint: Remember that the distance between two vectors f and g is the length of f − g.
c) Sketch x, cosx and sinx in the interval [0,1]. Give a geometric explanation of the numerical values found in parts a) –b)

Let the Fibonacci sequence be defined by F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 2.

Use induciton to prove that F0F1 + F1F2 + · · · + F2n−1 F2n = F^2 2n for all positive integer n.