Use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval. This question is from the differential equation.

y'-4y = x/y^2(y+1) , y(0) = 1; h=0.1, 0.05 , 0.025, on [0, 1]

13.6 Let G be a simple connected cubic plane graph, and let pk be the number of k-sided faces. By counting the number of vertices and edges of G, prove that

3p3 + 2p4 + p5 - c7 - 2p8 - • • • = 12.

Deduce that G has at least one face bounded by at most five edges.

A) Find the general solution of the given differential equation. y'' + 8y' + 16y = t−2e−4t, t > 0

B) Find the general solution of the given differential equation. y'' − 2y' + y = 9et / (1 + t2)

The first cake has three layers with each one having a different top view: the bottom is a square, the middle is a hexagon, and the top is a circle. The height of each layer may vary. The second cake has three layers with a top view of concentric circles. All three layers have the same height. The diameter of the middle layer is 20% larger than the top layer, and the diameter of the bottom layer is 25% larger than the middle layer.

Your data given is:

length of the side of cake1 square layer = 20

height of cake1 square layer = 1

length of the side of cake1 hexagon layer = 9

height of cake1 hexagon layer = 10

  diameter of cake1 circular layer =5

height of cake1 circular layer = 7

diameter of cake2 bottom layer = 68

The ratio of the volume of each layer to the volume of cake1 is 0.1514(square), 0.7966(hexagon), 0.052(cylinder)

Find:

1) The height of cake2

2) The total surface area of cake2

3) The volume of cake2

Answers should be: (I just don't know how toget there)

Second cake:
* total height: 1.047
* surface area: 3815.5796
* volume: 2641.8863

Suppose A and B are subsets of R, and define:
d(A,B) = inf{|a−b| : a ∈ A,b ∈ B}.

(a) Show that if A∩B 6= ∅, then d(A,B) = 0.

(b) If A is compact, B is closed, and A∩B = ∅, show d(A,B) > 0.

(c) Find 2 closed, disjoint subsets of R (say A and B) with d(A,B) = 0

Use variation of parameters to find a general solution to the following differential equations.

(a) y′′+y=tant, 0<t<π/2.

(b) y′′−2y′+y=et/(1+t2).

A periodic point of a function f : X → X is a point x ∈ X for which there is a number p (a period) with (f(x))^p = x (f^p denotes the composite (f ◦...◦f) of f with itself p times).

A fixpoint is a periodic point with minimal period p = 1, that is, a point x ∈ X such that f(x) = x.

For X = {1,2,...,n}, count the number of functions f : X → X such that each periodic point is a fixpoint.

Hint: What kind of vertebrate do you get when you apply Joyal's bijection on a function like this?

find the eigendata for A and B. (eigen value and eigen vector)

A = [4 1; 0 4]

B = [3 2 4; 2 0 2; 4 2 3]

Prove that the discrete topology on X is the same as the metric topology induced by the discrete metric.

Where metric topology is defined as:

If (X,d) is a metric space, then consider the collection T of all open subsets of X. Then (X,T) is topological space. This topology is called the metric topology on X induced by d.

Use Lagrange multipliers to find the minimum and maximum values for the following functions subject to the given constaints.

a) f(x,y) = 8x²+y² ; x⁴+y^{4 = 4}

b) f(x,y,z) = 2z-8x² ; 4x²+y²+z² = 1

c) f(x,y,z) = xyz ; x²+4y²+3z² = 36

Question 1

Problem 2.64 Use MATLAB to solve for and plot the response of the following models for 0≤t ≤1.5, where the input is f (t) =5t and the initial conditions are zero: 3¨ x +21˙ x +30x = f (t)

A)  Use math (analytically) to find the response of the models given in Problem 2.64 of your book. Show all assumptions and all steps of your derivation. (Turn in the calculations.)

B) Use MATLAB to solve the same problem 2.64

C) (Turn in the MATLAB script and answers from MATLAB,.m file, screen shots if needed)

E) Plot the response for Problem 2.64 . (Turn in the MATLAB plot with t being time in SI units)

F) Comment on the response the analytical solution compared with the MATLAB and the plots. (Do the calculations and MATLAB agree ?

Why and Why not.? Do the plots make sense?)

Which of the following are subrings of the field R of real numbers:

(d) D = {a + b (³√3) + c (³√9) | a, b, c ∈ Q}.

(e) E = {m + n(1 + √ 3)/2 | m, n ∈ Z}.

(f) F = {m + n(1 + √ 5)/2 | m, n ∈ Z}.

Discuss the ratio of circumference to diameter for circles on the sphere.

the Euclidean radius is the distance to the (Euclidean) center of the circle -- which
is not a point on the sphere. But the circle also has a center on the
sphere -- this would be the North (or South) Pole for lines of
latitude. So the spherical radius of such a circle is different, namely
the spherical distance from the (spherical) center.

Derive formulas relating these various notions of radius and circumference.