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**Unformatted text preview: **. Name: . . 391d F753 ........... ID number: .............................. April 20, 2009 ’ MATH 2271
' Differential Equations for Scientists and Engineers
Winter 2009 ' Second Midterm Exarn Time: 50 minutes 1. Consider the ﬁrst order differential equation y’- — \/ 2 + t2. {Lt-1 (3) Sketch the corresponding direction ﬁeld in the region 0 < t _<_ 1,
0 < y_ < 1. Kg 1 (b) Given that y( (0) = 1/2, use two steps of Euler’s method (with step ‘
size h: 1/2) to estimate y(1). LCJ” (7': 21(0)=)’ _ . :Y—(gﬂt): 83 rt
30:) a 31 = g, + WNW? 4 1:41 2. (a) Show that the functions 6‘ sint and 6‘ cost comprise a fundamen-
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Mil ﬁtﬂ’c’é: (Medial) 60166ch {a ‘lLe gm“ team/485k. Eéj (b) Find the general solution to the equation y” — 2y’ + 2y = sin t. W? Salt 4 ‘Paf'flow-(W whim N2 “(L #N%
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5f E1 1+2 Mum/+3975” : (A ~ZB)C067§ + (5 +20%” = SW; @ 92 :32: é> 3%; Aée‘gré‘ﬁh‘s‘ﬁl 3. Let yl and y2 denote solutions to the equation 3/” + e‘y’ + t4'g = 0. E5 I (a) If yl and y1 + 312 are linearly independent, what can you conclude
about the possible values of Wronskian W(y1,y2) of y1 and y2?
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' solution to the initial value problem yll + etyl+ t4y = 0, y(0) = 1, . yl(0) = 0
in terms of y1 and y2. We lulou/ llxe awwa/ $olw¥th I’M“ H“ 40"“
y: CIJMCZW‘ Q) o 4. Find the general solution to the equation 3/ ’+ y= sect.
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