20d_07s_1s - Math 20D 1. (a) second order linear (d) second...

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Unformatted text preview: Math 20D 1. (a) second order linear (d) second order nonlinear Solutions (b) first order linear (e) first order linear Friday 5/4/2007 (c) first order nonlinear 2. Compute y1 y2 − y1 y2 (the Wronskian) and see if it is nonzero at some point. Alternatively verify that one function is not a constant multiple of the other. 3. There are 3: y = −1 is stable, y = 0 is unstable and y = +1 is stable. Your answer should show how you got these results. 4. (a) The answer is 2 sin(3t). The general solution is c1 sin(3t) + c2 cos(3t). By the initial conditions, c1 = 2 and c2 = 0. Your solution should show how the general solution was found. (b) Separate variables: e−x dx = et dt and so −e−x = et + C . The initial condition gives −e−1 = 1 + C and so C = −1 − e−1 . Another way to write it: e−x + et = 1 + 1/e. (c) The equation is exact. Integrating gives x2 + xy − y 2 = C . (d) Divide by t to get it in standard form. An integrating factor is exp( −dt/t) = 1/t. Thus y /t − y/t2 = 1 and so (y/t) = 1. Solving, y/t = t + C . You could rewrite this as y = t2 + Ct. ...
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This note was uploaded on 06/18/2009 for the course MATH 20D taught by Professor Mohanty during the Fall '06 term at UCSD.

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