240-2006&2007-2-M10-March2007

240-2006&200 - Kuwait University Department of Mathematics 31 COmPUter Science Math-240 Midterm 1 Solve the following difierential

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Unformatted text preview: Kuwait University Department of Mathematics 31, COmPUter Science Math-240 Midterm 1 Solve the following difierential equations (Show your steps) (1) r _ - 2 ’9' my —:t:+y—:r:sm (w) x (2) ysezdx — (1 — 2yze“)dy = 0 (3) mg," + y = $2y2 1n($) (4) a y’ = —H~—ey y — 29:31:39”. (5). Describe the curve that is orthogonal to the family and passes through the point (0, “1). March 29th, 2007' Duration : 90 minutes 00 CO SOLUTIONS Please note that each question is worth m points. 1. y’ = 1+ y/a: — sin2 y/ac .. put to = y/m so that dy = wdm + mdw .. equation reduces to malts/dz = 1 — sin2 to = c052 u; .. family of solutions In —- tan(y/s;) = C. 2. yaexdm— [1—2yze”) dy = [l .. put to = em and you‘ll have y3dw+(2wy2 - lldy = 0.. take M = 3?, N :2wy2—1.. My = 3y2, Nw = 2y2, and (My —Nw)/M = 1/1; .. we have u = exp(— f 1/3; dy) = 1/3; .. multiply by a to get the exact equation y2dw + (2mg —1/y)dy = 0 . Fw = y2 gives F = wyz + 9(y) .. now F3, = 2m; + g'(y) = 2103; — 1/y and it follows that g[y) : —ln|y[ + Cl and F = wyz — lnlyl — 01 = 02 .. family of solutions becomes emy2 - lnly| = C. 3. xy’ + y = By? 1111: .. or in plain English wdy + yda‘ = d(:z:y) = (mmzln :cdx .. separate variables to get d(xy]/(:cy)2 = lnzolz .. hence x (11139 - 1) +1/(Icy) = C'. 4. eygdx + (2sye’5‘2 — y)dy = 0 is an exact equation .. Fz : egg gives F = self + 9(y) .. Pg 2 Zayey2 +g’(y) 2 2159's”: —y .. so 9(y] = —y2/2+ 01 and F = 2:332 — y2/2 + C1: Cg .. family of solutions becmnes 2mg: — yz : C. .\_ _ ..... d 5. f(:c,y,c) = y —Cm4 = 0.. c = y/scd‘ and (sly/dash - 4&3 = 0.. (sly/day : dig/5c and for the g-family we have (dy/dm)g = —x/(4y) or 4ydy + a‘dm = 0 .. hence 2y2 + 2:2/2 = k .. the particular g-curve passing through (0, —1) is the ellipse 1:2 —I— 19/4 : 1. OO 99 ...
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This note was uploaded on 02/23/2010 for the course DIFFRENTIA 0410244 taught by Professor Diffrential during the Spring '10 term at Kuwait University.

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240-2006&200 - Kuwait University Department of Mathematics 31 COmPUter Science Math-240 Midterm 1 Solve the following difierential

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