solutionofexam_1(MTH205)

solutionofexam_1(MTH205) - Existence and unigueness theorem...

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Unformatted text preview: Existence and unigueness theorem and Direction Fields Q1. (8 points) a)Determine the region ofthe xy~plane for which the initial value problem , 1ly+l 1; 2m , y(x0):yg X we 2%: {3' t h) The following graph represents the direction field for the differential equation y'=\/}7 Sketch approximate solution curves through the point (0,1) We, «(‘4 -C~f~g “erz'w. «4*; «.‘w '.~x- "van-u" » ‘~ , .‘ r ” “ t ~<~ r~t \. -. a —~ 1 u. w "x \4 a e 'w ‘u. N o) Solve the differential equation in part (b) and find the solution that satisfies y(0) : l . Find the largest interval I on which the solution is defined. Autanomous equation and Qhase portrait Q2. (5 points) Consider the differential equation #3 2y ‘ V" . a) Find the equihbrium selutions and phase portrait. e) Expiain why the solution through the point (0,1) do not intersect the line y = 2 . Sketch approximate sohltion curve through the point. 3 Methods for solvin first order differential e nations Q3. (8 points) Show that the differential equation: [sin x - e” + fie“ kl): + [2y + xe‘" + 2329‘ My 2 0 is exact. Find the soluiian that satisfies the initia! condition y(0) = 0. X ('3 C08 3: V tan )1? Z V i and solve the differentiai equations: )’ " .1 Classifv Q4. (8 points) NW_W~,.A.MW, ,, Q5. (8 points) Use an appropriate substitution method to solve: (2x + y + By” :1 , , we)»: ()6 (8 points) Show that the dit‘f‘ercntiai equation: x 1110) / flafy : [)2 In(y / x) ~ fldx is hemogcncous and 501w: it. , A N x V1 : g V » <7 i /; V, y i \ V" g f/ Q “1" d j 7 \‘r (w my a“ i y W ‘ “2 : % E X f /, 1‘ am 12% ’ z' i y/ E g ,» w \ <' } )3 xf W m § § '2 ‘1» {g gawk m i b E V; g 28‘; m V g k} 1 ! f 3? 3W3” : >0 “‘ “2W: i; g ty’a w} h 74 ah? '3 :55 >< V‘é ‘ x E ’9“ k1 x 6 A ) )lications of first order differential e nations Q7 (8 points) A tank initialfy contains 1 20 Iiters oi‘purc: water. A mixture containing a coaccntration 0f‘80 gramsflimr ofsak mm” the tank at rate of 2 Htersfmin, and the “CH—stirred mixture 1czwe3 the tank at the Same rate. a) LctQU) be the amount ofsak in the tank at time Min minutes). Set up the WP t0 modci the variation in the amoum ofsait in the tank, 2/” a” i: x: m w“ 53 I“ A a g. * g A: ...
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