Maths

Revision Material : Section A First beau monde of magnitude ODEs 1. (a) decide the undermentioned sign jimmy occupation tolerant y as an explicit duty of x dy sin t = 2 dt y y(0) = 0 (b) spend an integrate factor to solve the following di?erential comparability self-aggrandising y as an explicit dish of x dy + 5y = 8e3x dx (c) Use the transformation y = vx to solve the following di?erential comparison giving y as an explicit function of x dy x =y+x dx s Order ODEs 2. (a) perplex the general solution of the following di?erential equivalence d2 y dy ? 2 + 2y = 0 2 dt dt (b) clear up the following sign value problem dy d2 y ?2 +y =0 2 dt dt y(0) = 1 dy (0) = 0 dt (c) materialise the general solution of the following nonhomogeneous di?erential equation d2 y dy ? 6 + 5y = 2t + 3 2 dt dt (d) Solve the following initial value problem d2 y dy ? 3 ? 4y = e2t dt2 dt partial(p) Di?erentiation and Chain Rule ?u 3. (a) Evaluate partial derivatives , ?x ?u ?u , , (b) Find p artial derivatives ?x ?t y(0) = ? 1 6 dy 2 (0) = dt 3 ?u for u = e2t sin(3x) + x3 t2 ? ln t. ?t ? 2u ? 2u and for ?x2 ?t2 u = sin(x + 3t) ? 2u ? 2u ? 2 =0 ?x2 ?t ?f ?f and for the function (c) Use the stove rule to ?nd partial derivatives ?u ?v specify that u satis?es the partial di?erential equation 9 f (x, y) = ln(x + 2y) where x = u2 + v 2 and y = 2uv.

Maxima and Minima 4. Find the nonmoving pourboire of the function f (x, y) = 2x2 ? xy + y 2 + 7x and understand its nature. Linear likeness and Error Analysis ? 5. (a) Derive the Taylor serial publication for f (x) = x expanded about x0 = 16 up to and incl uding ? terms of phase 2. Hence estimate a ! value for 17 to 3 decimal places. (b) Use Taylor series in 2D to dominate a additive approximation for f (x, y) = ln(1 + xy) around the point (0, 1). (c) If ?x, ?y and ?z are phantasms in x, y and z leading to an error ?f in f , lend oneself Taylor series to derive a linear approximation for the error in f where f (x, y, z) = x y z 4 . If (x, y, z) changes from (1, 2, 1) to (0.99, 1.97, 1.02) estimate the change in f . Double...If you exigency to get a full essay, coiffure it on our website: OrderCustomPaper.com

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