Find the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

3y"-2e^6xy'+4(cosx)y=0; y(0)=1, y'(0)=1

Type an expression that includes all terms up to order 3.

Use a Laplace transform to solve the initial value problem: y''' =y+1,

y(0) = 0,

y'(0) = 0,

y''(0) = 0.

The current spot price of a stock is $33.00, the expected rate of return is 7.2%, and the volatility of the stock is 20%. The risk-free rate is 3.8%.

(a) Find the 90%-confidence interval for the stock price in 6 months.

(b) Compute the expected percent change in the stock over the next 6 months.

Solve 3xy'' + y' - y = 0 by the Method of Frobenius.

Define
1). homogeneous function
2). Euler's theorem
3). le chatelier's principle
4). young's theory

Let A and B be finite sets. Prove the following:

(a) |A∪B|=|A|+|B|−|A∩B|

(b) |A × B| = |A||B|

(c) |{f : A → B}| = |B||A|

Prove for the following:

a. Theorem: (Cantor-Schroder-Bernstein in the 1800s) For any set S, |S| < |P(S)|.

b. Proposition N×N is countable.

c. Theorem: (Cantor 1873) Q is countable. (Hint: Similar. Prove for positive rationals first. Then just a union.)

Exercise We say that |A| = |B| if there exists a bijection f : A → B.

(a) Show that the intervals [1, 2] and [3, 7] have the same cardinality.

(b) Show that the intervals (0, 1) and (0, ∞) have the same cardinality. (Hint: First try (0, 1) and (1, ∞) (later subtract 1). Identify numbers small numbers in (0, 1) with large numbers in (1, ∞) somehow.)

(c) Show that the interval (0, 1) and the real numbers R have the same cardinality. (Hint: tan^-1x.)

What has been the contribution of this history of mathematics course to your understanding of mathematics, if any? Please provide specific details.

How might the history of mathematics be important to the way we practice mathematics?

Power series

Find the particular solution of the differential equation: (x^2+1)y"+xy'-4y=0 given the boundary conditions x=0, y=1 and y'=1. Use only the 7th degree term of the solution. Solve for y at x=2. Write your answer in whole number.

Block copy, and paste, the argument into the window below, and do a proof to prove that the argument is valid.

1. (p   ⊃ z)   •   (q • ~z)
2. (~p • q)    ⊃   s
3. m   v ~s     : .    m

NO WIKI or PLAGIARIZING!!

Reflect on the concepts of linear and non-linear systems. What concepts (only the names) did you need to accommodate the concept of linear and non-linear systems in your mind? What are the simplest linear system and non-linear system you can imagine? In your day to day, is there any occurring fact that can be interpreted as linear systems and non-linear systems? What strategy are you using to get the graph of linear systems and non-linear systems?