Use the Runge-Kutta method with step sizes h = 0.1, to find approximate values of the solution of

y' + (1/x)y = (7/x^2) + 3 , y(1) = 3/2 at x = 0.5 .

And compare it to thee approximate value of y = (7lnx)/x + 3x/2

Consider a general system of linear equations with m equations in n variables, called system I. Let system II be the system obtained from system I by multiplying equation i by a nonzero real number c. Prove that system I and system II are equivalent.

(A) Using a straight edge, sketch the [1̅23̅] direction within a cubic unit cell and label start and end points as well as x, y, and z axes. (B) Using a straight edge, sketch the (1̅23̅) plane within a cubic unit cell and label x, y, and z axes as well as the x, y, z axial intercepts. (C) What is the geometric relationship between the (1̅23̅) plane and the [1̅23̅] direction?

solve the SDE dX = − 1 /τ( (X − µ)dt) + (2D)^-1/2 dB. X(t = 0) = β

Example 3.5: Again let X = Y = R. Define g by g(x) = x2. The graph of this function has the familiar parabolic shape as in Figure 3.1(b). Then for example, g([0, 1]) = [0, 1], g([1, 2]) = [1, 4], g({−1, 1}) = {1}, g−1([0, 1]) = [−1, 1], g−1([1, 2]) = [− √ 2, −1]∪[1, √ 2], g−1([0, ∞)) = R.

*I need help understanding why each example in bold is the answer it is*

*Please explain clearly why the inverse functions have the answer they have because it is not clear to me why*

*Please show all work and step by step solution*

Let G be a group. (consider the following parts that go together):

(1) Prove that (a^-1ba)ⁿ = a^-1bⁿa for any a,b in G, and any integer n.

(2) Prove that |xax^-1| = |a| for any a, x in G.

(3) Prove: If a is the only element of order two in G, then a lies in Z(G) where Z is the center of the group, G.

Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 25 grams of A and 53 grams of B, and for each gram of B, 1.2 grams of A is used. It has been observed that 19.5 grams of C is formed in 15 minutes. How much is formed in 30 minutes? What is the limiting amount of C after a long time ?

_____ grams of C are formed in 30 minutes

_____ grams is the limiting amount of C after a long time

10. A student has to take eight hours of classes a week. M-F. They want to have fewer hours on Friday than on Thursday. In how many ways can they do this? Assume that student need not go class everyday, just need more hours on Th than Fr.

A 1 kg mammal uses about 5 kcal of energy per hour (metabolic rate). It eats hay, which contain about 2000 kcal per kg.

a. Write out the equation to calculate how much energy it uses per week:

b. Write out the equation to determine how many kg of hay this mammal must eat per in a week to maintain its body weight

c. Write out the equation to determine how long it will take this mammal to burn 0.1 kg of fat, assuming that fat yields 9 kcal/gram

2. Give an example of a linear transformation L : lR^2 arrow lR^2 which has a repeated real eigenvalue, but in which lR^2 does posses a basis of eigenvectors.

Suppose f : R → R is measurable and g : R → R is monotone. Prove that g ◦ f is measurable

Developing filters using convolution theorem and Fourier Transform.

You have been hired as an Engineering Mathematician at a consulting firm located in Saint Louis. On your first job, you have been asked to mathematically design a frequency filter that removes from a standard beacon signal a periodic interference generated by a rotatory machine located in the basement of the company. Please see below for more details: Let s(t) be the standard beacon signal that is being communicated. Below you can find its Fourier Series representation ?(?)=2.5+2 sin(?)+3cos(t)+0.5cos(2?)+ 0.3 sin (2?). The periodic interference is given by ?(?)=0.5 cos (120 ?) and the measurement signal with noise is given as follows: ?(?)=?(?)+?(?) Let g( t) be the filter function and let ? Z(?) be the function that results from applying ? (?) to ?(?). Using the Fourier transform of the convolution theorem, propose the design of a filtering function g(t) which removes from ?(?) the effect of the periodic noise ?(?) assuming that we only know that its fundamental period is equal to 2pi/120. Make sure to write the analytical expression of ?(?) and its Fourier transform. Also, please write the mathematical expression that relates ? Z(?) as a function of ?(?)and ?(?)

How many eight-digit positive integers have the sum of digits being even?

Let P(n) := ” If n^3 is odd then n is also odd.” I.e., if ∃k ∈ Z, n3 = 2k + 1, ∃b ∈ Z, n = 2b + 1

a) Prove P(n) by contraposition

b) Prove P(n) contradiction

c) Prove P(n) using induction