Solve the following initial value problem: y'''-4y''+20y'=-102e^3x, y(0)=3, y'(0)=-2, y''(0)=-2

Find the first five nonzero terms in the solution of the given initial value problem. y′′−xy′−y=0, y(0)=7, y′(0)=10

Find the first five nonzero terms in the solution of the given initial value problem.

y′′+xy′+2y=0, y(0)=8, y′(0)=9

Show an example of a system which is a quasigroup butnot a group.

Please provide explanation.

Each of the following functions has a critical point at the origin. Show that the second derivative test fails there. Determine whether the functions has a local maximum, local minimum, or saddle point at the origin by visualizing what the surface z=f(x, y) looks like. Describe your reasoning.

(a)f(x, y) =x^2y^2

(b)f(x, y) = 1−xy^2

research and  find an important mathematician or scientist whose work is especially interesting or important.?

Explain briefly why their discoveries are important for us today?

Can you find any particular analytical trait that would be advantageous for us to have?

Solve the following initial value problem using the undetermined coefficient technique:

y'' - 4y = sin(x), y(0) = 4, y'(0) = 3

Solve the given initial value problem.

y'''+2y''-13y'+10y=0

y(0)=4    y'(0)=42 y''(0)= -134

y(x)=

Let A, B, C be arbitrary sets. Prove or find a counterexample to each of the following statements:

(b) A ⊆ B ⇔ A ⊕ B ⊆ B

Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*. Can someone please explain this to me? I get that f(1) = 0, f(2) = -1 but then f(-1) = -f(1) = 0 and f(2) = -f(2) = 1 but this does not make sense in order to define a function. Can someone explain why this is not right and show what it is correct?

Let f:A→B and g:B→C be maps.

(a) If f and g are both one-to-one functions, show that g ◦ f is one-to-one.

(b) If g◦f is onto, show that g is onto.

(c) If g ◦ f is one-to-one, show that f is one-to-one.

(d) If g ◦ f is one-to-one and f is onto, show that g is one-to-one.

(e) If g ◦ f is onto and g is one-to-one, show that f is onto.

1. Prove that Z[√3i]= a+b√3i : a,b∈Z is an integral domain. What are its units?

Euler’s method

Consider the initial-value problem y′ = −2y, y(0) = 1. The analytic solution is y(x) = e−2x . (a) Approximate y(0.1) using one step of Euler’s method. (b) Find a bound for the local truncation error in y1 . (c) Compare the error in y1 with your error bound. (d) Approximate y(0.1) using two steps of Euler’s method. (e) Verify that the global truncation error for Euler’s method is O(h) by comparing the errors in parts (a) and (d).

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for the transition matrix P= 0.8 0.2 0.0 , solve the equation SP=S to find the stationary matrix S and the limiting matrix P.

0.5 0.1 0.4

0.0 0.6 0.4

discrete structures problems

1.Find a limit to show that x(In(x²))³ is O(x²). Simplify when possible to avoid doing more work than you have to. You will need to use L'Hôpital's rule at least once.

2.Suppose that f is o(g). What is lim(f(n)/g(n)) as n→ ∞?

3.Suppose that algorithm has run-time proportional to log n and takes 1 millisecond to process an array of size 3,000. How many milliseconds will it take to process an array of size 27,000,000,000 ? Hint: what simple relationship is there between the first number and the second number?