Find Laplace Transforms - Mathematics Assignment Help

Assignment Task

1. Determine, with justification, whether the following series converges or diverges.

2. Determine values of x for which the following series converges and find its sum. Do NOT simplify the sum.

3. Find the Taylor series about x = a, where a > 0 is a constant, for the function f(x) = sinx. Write your final answer in sigma notation, simplified as much as possible. You must use a method that guarantees that the series converges to f(x).

4. Find the Taylor series about x = ?3 for the function

Write your final answer in sigma notation, simplified as much as possible. You must use a method that guarantees that the series converges to f(x). What is the interval of convergence of the series? Hint: Do not divide x + 3 by x + 5.

5. Solve the initial-value problem

6. Three chemicals A, B, and C combine to produce chemical D. Three grams of A, two grams of B, and one gram of C make 6 grams of D. The rate at which D is formed is proportional to the product of the amounts of A, B, and C in the mixture. If 100 grams of A, 20 grams of B, and 50 grams of C are brought together at time t = 0, set up, but do NOT solve, an initial-value problem for the number of grams of D in the mixture at any given time. Determine the amount of D in the mixture when reactions finish.

7. The differential equation is linear, but it does not have constant coefficients.
Show that a general solution of the associated homogeneous equation can be obtained bysetting y = xm and finding values for m.
Find a particular solution of the nonhomogeneous equation by setting yp(x) = Ax3, and finding A.
What is a general solution of the nonhomogeneous equation?
For what values of x is your solution valid?

8. Find Laplace transforms for the following functions:
f(t) = cos(4t + 1)h(t ? 1)(b)f(t) = 4 ? t2,0 < t < 2,f(t + 2) = f(t)

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