How do we find singular solutions for a first order Differential Equation?

Using Picard’s Uniqueness Theorem or otherwise, prove the Existence and Uniqueness Theorem for a First Order Linear Differential Equation.

Find the general solution of y 00 + y = (tan t) 2 .

y''+y=2t+1+(cos(t))^-2

Each of the faces of a regular tetrahedron can be painted either black or
white. Up to rotation, how many ways can the tetrahedron be painted? Please provide
all the necessary computations and explanations. (hint : idea of abstract algebra can be used)

solve the edo below by laplace transform

A metal plate is 50 cm long, at the end x = 0 to
plate temperature is = 980K. The one-dimensional temperature distribution on the plate
metal is given by the following EDO,d^2T/dx=144T+12 Knowing that T '(0) = 0. Solve for
Laplace and find the algebraic equation that represents Temperature as a function of
plate length X.

answer

T(x) = -1/12 + 490,04 e^-12x + 490,04 e^12x

The mathematical model describes the behavior of a pendulum and is
represented by:

0,5 dQ/dt=24sen(t)+15

One person holds the pendulum and releases it, Q = 0 at t = 0s. Describe this behavior of the pendulum finding the amplitude equation as a function of time. Sketch the graph. On what height is the pendulum when t = 7s?

answer:Q(t) = 48(1 – cost) + 30t

Real Analysis:

Detail the constructed definition of the Riemann Integral and discuss conditions for the existence of the integral.

Show how to reduce 3-SAT problem to 3D-Matching problem.

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉.

Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

Let A =   [  0 2 0

1 0 2

0 1 0 ]  .

(a) Find the eigenvalues of A and bases of the corresponding eigenspaces.

(b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line.

(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist.

(d) Write down explicitly a diagonalizing matrix S, and a diagonal matrix Λ such that S −1AS = Λ; A = SΛS −1 . or explain why A is not diagonalizable.

Nonlinear Dynamics and Chaos problem 10.7.10

Fill in the missing algebraic steps in the concrete renormalization calculation for period doubling.Let f(x) = -(1 + mui)x+x^2. Expand p+(n)subn+1 =

f^2(p+(n)subnetting) in power of small deviations (n) subn using the fact that p is a fixed point of f^2.

Prove that there exists a negative number.

The algorithm is basically as follows. The notation is slightly different from that in the website you were given, but there is no difference in the method.

Given the initial value problem

dy/dx=f(x,y),y(a)= y_0

Euler’s Method with step size h consists in applying the iterative formula

y_(n+1)= y_n+h∙f(x_n,y_n ),n≥0

To compute successive approximations y_1,y_2,y_3,⋯ to the (true) values 〖y(x〗_1),〖y(x〗_2),〖y(x〗_3),⋯ of the exact solution y=y(x) at the points x_1,x_2,x_3,⋯, respectively.

In plain English:
You want to approximate the value of dy/dx (or y’) at some point in an interval.

Step 1: Depending on how accurate you need to be, divide the interval up into little pieces of equal length; this length is the step size h. For purposes of discussion, let’s use the interval [0,1] and use ten intervals, so h = 0.1.

Step 2: y_0=0
Step 3: y_1=y_0+0.1f(x_0,y_0)
Step 4: y_2=y_1+0.1f(x_1,y_1)
…
Stop after ten steps, in this case. Usually the stopping criterion is a level of accuracy.

You can easily set this up in Excel.

Exercises
Use Euler’s Method with step sizes h =0.1,0.02, 0.004, 0.0008 (that is, do the problem 4 times, each with a more precise value of h) , 10 equally spaced iterations.


1. y^'=x^2+y^2,y(0)=0,0≤x≤1

2. y^'=x^2-y^2,y(0)=1,0≤x≤2

3. y^'=ln⁡y,y(1)=2,1≤x≤2

4. y^'=x^(2/3)+y^(2/3),y(0)=1,0≤x≤2

5. y^'=x+√x,y(0)=1,0≤x≤2

6. y^'=x+∛x,y(0)= -1,0≤x≤2

On June 24, 1948, the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The Americans had two types of planes available, the C-47 Skytrain and the C-54 Skymaster. The carrying capacity was 3.5 tons for a C-47 and 10 tons for a C-54. To break the Soviet blockade, the Western Allies had to maximize carrying capacity, but the Americans were limited by the following restrictions:

- No more than 44 planes could be used per day

- Each C-47 required 4 crew members per flight and the crew requirement for the C-54 was 5. The total number of personnel available per day could not exceed 200.

- The American's only had 32 C-54's Available.