Unformatted text preview: Math 20D Second Midterm 48 points May 26, 2004 0 Print Name> ID number and Section on your blue book. 0 BOOKS and CALCULATORS are NOT allowed.
One side of one page of NOTES is allowed, 0 You must show your work to receive credit. Some integrals and derivatives that may be useful: ftancr dm = iln(cos Cc) + C (tancr)’ = sec2 :r (seem)’ = sec a tanm
fsecm dm : ln(sec m + tan m) + C (arctan m)’ : #5 (arcsinm)’ : 1; 1. (27 points) Find the general solutions of the following differential equations
(a) @563”)? :1 (b) fizz/+9 : 44 (C) (m2+1)9’*y : 0 1. (a) An integrating factor is exp (f —tanm dz) : cos 1. Thus (3/ cos a)’ : cosm and 1 ' C
9(1) : fcosmdr : $ : tanm+0secm
cosm cosm (13) Since the characteristic equation for the homogeneous equation is 0 : r2—2r+1 : (r—IY,
y : Gist + Ogtet is the general solution to the homogeneous equation. By un
determined coefﬁcients, a particular solution is y = At + B. Since 3;” = A and
y” = 0) we have 4: = 072A+CAt+Bj = At+CB72Al Thus A : 4 and B : 8, The general solution to (b) is therefore y : Clef + my + 47: +8 (c) Rearrange, separate variables and integrate: is
a dm
: /1+m2 and so lny : arctanm+O You may leave the answer this way, with or without absolute values on y, or you
may solve for y. 2. (3 points) I have decided to ﬁnd a series solution 3/ : Z anm” to
(4 + mg” + (1 7 my : 0. For what values of m can you guarantee that the series will converge? Why?
You must give a reason—the ‘Why?”—to receive credit. 2, Since 4 + 032 = U for o: = :21, the radius of convergence of the series for 41%;; is 2,
Thus the best we can guarantee is lml < 2. 3. (9 points) 3/ = t and y = t2 are solutions to $2912, 2th + 23/ = 0. Find a particular
solution to tgy'” 7 2:3/ + 23} : tzef. You may leave integrals in your answer, 2
3, Since 71’ e 2y“/t+y/t2 = 8* and WW2] = li ta: $28t 2 tat s 2 start
y:—tft—2dt+tft—2dt:—t/edt+tft 4, (9 points) Mt) satisﬁes the differential equation = t2, a particular solution is 2/(i)+y(t1):1,
with y(t) : U for t S 0, Compute the Laplace transform Y(s) : £{y(t)}. 4, By the table for Laplace transforms, 1 sY(s) +y[0)+e’5Y(sj = 1/5 and so Y(s) = ml When your exam is returned, it will have the grade you will receive if you do NOT
talre the ﬁnal exam. In computing that grade) this exam will be weighted more than the
ﬁrst exam to reﬂect the fact that the ﬁnal will emphasize differential equations. END OF EXAM ...
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 Spring '06
 Mohanty

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