sol1 - MAP 2302 Fall-2011 Section 0100 Quiz 1 1. Determine...

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Unformatted text preview: MAP 2302 Fall-2011 Section 0100 Quiz 1 1. Determine whether the differential equation ds = t ln(s2t ) + 8t2 dt is separable. Justify your answer. SLUTION: It is separable, since it can be written as ds = t2 (ln s2 + 8) dt in view of the fact that ln s2t = 2t ln s = t ln s2 . 2. Solve the initial value problem sin x dy + y cos x = x sin x, dx y (π/2) = 2. SOLUTION. This a linear equation with P (x) = cos x sin x and Q(x) = x in the standard form. One can find µ and multiply the equation to get the derivative of the product on the left. If one is really smart, he/she can notice that we already have the derivative of the product on the left: d (y dx dy sin x) = sin x dx + y cos x. Thus, our equation is d (y dx sin x) = x sin x. Playing with µ would give the same result. We itegrate both sides to obtain y sin x = gral can be evaluated by parts: x sin xdx = x sin xdx + C . This intexd(− cos x) = x(− cos x) − (−cosx)dx = sin x − x cos x. Thus, y = 1 + C −x cos x . sin x C −x×0 For x = π/2 we obtain 2 = y (π/2) = 1+ 1 ⇔ The ANSWER: y =1+ 1 − x cos x . sin x 1 2 = 1+ C ⇔ C = 1. ...
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This note was uploaded on 12/15/2011 for the course MAP 2302 taught by Professor Tuncer during the Spring '08 term at University of Florida.

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