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Unformatted text preview: MAP 2302 Fall2011 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.
 Spring '08
 TUNCER

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