5. (x2 - 3)y" + 2xy' = 0

5. Consider the equation f(x) = x – 2- = 0. (a) Show that the above equation has a solution on the interval [0, 1]. (b) Prove that the sequence rk+1 = 2¬** converges to the solution.

(x-2) √x²+2x x-2) Converges? Dieverges? dx AND WHY?

Prove that for integers y=1 the equation 3 h-n (2 x ₁ + x₂ + .. + x₂) +3²(1)(2x₁+x₂+.+xn-₁) +...+ 3^-1 (2²₁) Y = 2x₁+x₂+...+% -3" cannot be solved with positive integers X₁₁X₂... Xny no

Find two linearly independent solutions y₁(x) and y₁(x) to y"-2xy' -2y = 0 about the ordinary point x=0. You DON'T have to express the solution via a general formula involving summation but do write out the first four nonzero terms of y₁(x) and y2(x). STATE THE RECURRENCE RELATION.

Show all steps and work out problems completely please

4. Suppose that a single bacterium is in a bottle at 11:00 am. It divides into two bacteria at 11:01 am, and the population continues to double every minute until the bottle is completely full at 12:00 noon. Find how many bacteria there are at 11:26am. Show work or calculations! (4.1). (4.2) Work: 11:00-11.26 = 26 min + = 26 = 20 = 67,108,864 = 11:26 am Answer: 11:26 am = 67,108,864 Find the fraction of the bottle that is full at 11:55 am. (Hint, work backward!) Show calculations or give an explanation. Work: Answer:

x²+1 * 11. Find x'+3x+1

Show that coth-1=1/2ln(1+x/x-1

4. Find the first three nonzero terms of the Taylor series of the function f (x) = sin x in the powers of x - 1 6 sin x = 글-블 (-증)- 1(x-풍)' + sin x 글 + 플(-증) -1(+-증)' = - +... 2 None of them 글 + 블(x-종) + 1(x- 증)' + . sin x = sin x = (x-) - (x-)' + -- +...