20d_04s_2s

20d_04s_2s - Math 20D Second Midterm Solutions 1(a An...

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Math 20D Second Midterm Solutions May 26, 2004 1. (a) An integrating factor is exp (integraltext - tan x dx ) = cos x . Thus ( y cos x ) = cos x and y ( x ) = 1 cos x integraldisplay cos x dx = sin x + C cos x = tan x + C sec x. (b) Since the characteristic equation for the homogeneous equation is 0 = r 2 - 2 r + 1 = ( r - 1) 2 , y = C 1 e t + C 2 te t is the general solution to the homogeneous equation. By un- determined coefficients, a particular solution is y = At + B . Since y = A and y ′′ = 0, we have 4 t = 0 - 2 A + ( At + B ) = At + ( B - 2 A ) . Thus A = 4 and B = 8. The general solution to (b) is therefore y = C 1 e t + C 2 te t + 4 t + 8 . (c) Rearrange, separate variables and integrate: integraldisplay dy y = integraldisplay dx 1 + x 2 and so ln y = arctan x + C. You may leave the answer this way, with or without absolute values on y , or you may solve for y . 2. Since 4 +
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