Chapter 5RoomB. Similarly, we obtainy0=±1000·21000+12(x−y)²−15(y−0) = 2 +12x−710y.Hence, the system governing the temperature exchange isx0=20−(3/4)x+(1/2)y,y0=2+(1/2)x−(7/10)We fnd the critical points oF this system by solving20−(3/4)x/2)y=0,2+(1/2)x−(7/10)y⇒3x−2y=80,−5x+7y=20⇒x= 600/11,y= 460/11.ThereFore, (600/11,460/11) is the only critical point oF the system. Analyzing the directionfeld, we conclude that (600/11,460/11) is an asymptotically stable node. Hence,limt→∞y(t)=46011≈41.8◦±.(One can also fnd an explicit solutiony(t) = 460/11 +c1er1t+c2er2t,wherer1<0,r2<0,to conclude thaty(t)→460/11 ast→∞.)39.Letybe an arbitrary Function di²erentiable as many times as necessary. Note that, For adi²erential operator, say,A,A[y
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.