In the given Initial Value Problem, y" – 27y = f (t); y (0) = y' (0) = y" (0) = 0 with f (t) = { for 0 5 What would be the correct Partial Fraction Decomposition? Ο Υ (s) -Α+ Bs+C s3-27 Ο Y(s) : Ds+E + s2+38+9 A B C 8-3 s2 +3s+9 C(3s²) 'Y (s) = 4 + B s3-27 53 -27 D(2s+3) O (s) = + B C 8-3 s2 +3s+9 s2+3s+9 +

3x-11 dr - partial fraction 1. J 212 -x-6 |

Eliminate the arbitrary constants

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= 12. (a) Find the unique quadratic polynomial y ax²+bx+c that passes through the three points (1, 1), (3,5), and (-2, 0) in the x-y plane. (Hint: this will involve a linear system where a, b, and c are the unknowns.) (b) How many data points (x, y) should be needed to uniquely determine the coefficients of an nth degree polynomial anx" +an-1² -1xn-1+...+ a₁x+ao? Explain your answer in terms of linear systems.

Hi, can you help me with all parts of this question. thank you

x* -1 (b) If f(x): decompose f(x) into its partial fraction to evaluate [ f(x) dr. x +x +X

5а2 — 6х — 2 The partial fraction decomposition of can be written in the form of (x – 2)(x2 – 3x + 5) f(x) (x – 2) g(x) (22 — За + 5)' where - f(æ) = g(x) =

If y = (1 + x)(1 + x²)(1 + x*)....(1 + x* ) then d at x = 0 is -

17. Find ø if (a) Vø = (y² – 2xyz³)i+ (3+ 2xy – x²2³)j+ (4z³ – 3x²yz²)k and ø(1, 1, – 1) = 4. (b) Vø = 2xyz³i+x²z³j+ 3x²yz²k and ø(1, –2, 2) = 4.