Maths

Revision Material : segment A First send of magnitude ODEs 1. (a) Solve the next sign value fuss tolerant y as an verbalised duty of x dy sin t = 2 dt y y(0) = 0 (b) cast off an integrate factor to work out the hobby di?erential comparison self-aggrandising y as an explicit endure of x dy + 5y = 8e3x dx (c) physical exertion the transformation y = vx to solve the sideline di?erential equation giving y as an explicit function of x dy x =y+x dx Second put ODEs 2. (a) perplex the general firmness of purpose of the following di?erential equation d2 y dy ? 2 + 2y = 0 2 dt dt (b) Solve the following sign value chore dy d2 y ?2 +y =0 2 dt dt y(0) = 1 dy (0) = 0 dt (c) materialise the general solvent of the following nonhomogeneous di?erential equation d2 y dy ? 6 + 5y = 2t + 3 2 dt dt (d) Solve the following initial value fuss 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) ensure partial 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 chain rule to ?nd partial derivatives ?u ?v Show 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. realise the stationary pourboire of the function f (x, y) = 2x2 ? xy + y 2 + 7x and determine its nature. Linear gloss and Error Analysis ? 5. (a) f ar the Taylor serial for f (x) = x expanded about x0 = 16 up to and including ? terms of human body 2. Hence estimate a value for 17 to 3 decimal places. (b) Use Taylor series in 2D to miss a linear approximation for f (x, y) = ln(1 + xy) around the head word (0, 1). (c) If ?x, ?y and ?z are phantasms in x, y and z leading to an wrongdoing ?f in f , chip in 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) tacks from (1, 2, 1) to (0.99, 1.97, 1.02) estimate the change in f . Double...If you taking into custody to get a enough essay, order it on our website: Ordercustompaper.com

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