Find the maclaurin series

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Reference no: EM133056222

Question 1: Solve the differential equation,

(1 + cos(x) + 2sin(x)) dy/dx + (sin(x)) - 2cos(x))y = 4sin(x) - 8cos(x).

Question 2:

(a) By using the Maclaurin series for ln(1 + x) or otherwise, find the Maclaurin series for ln (1/1 - 3x³) up to and including the term in x¹².
(b) Hence find,

lim _x->0 In (1-1/(1-3x²)/ x³

Question 3:
(a) Use the binomial theorem to expand (cos(θ) + isin(θ))³. Give your answer in the form of a +bi.

(b) Use de Moivre's theorem and the result from (a) to show that cos(3θ) ≡ 4cos³(θ) - 3cos(θ).
Let w be one of the non-real solutions of the equation z⁷ = 1.

(d) Hence find the value of cos (2Π/7) + cos (4Π/7) + cos (6Π/7).

Mr. Pettifor solves the equation 8m³ + 4m² - 4m - 1 = 0 using his calculator and finds the following solutions rounded to three significant figures.

m = -0.901

m = -0.223

m = 0.623.

(e) Find cos (2Π/7).

Reference no: EM133056222

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