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Instructions: If your homework is not already in electronic format, please scan it into one PDF file, and upload it on GradeScope

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Instructions: If your homework is not already in electronic format, please scan it into one PDF file, and
upload it on GradeScope. Possible apps for scanning your work are CamScanner or GeniusScan: if you
believe that turning your document into Black & White (not grayscale!) makes it more readable, please
do so (also: writing in pen instead of pencil may make the scanned document more readable). GradeScope
makes it possible to ‘tag’ the individual problems within your PDF file: please do it in order to make the
graders’ work a bit simpler. Please solve the problems in the given order, and show your work.
Problem 11.1 (Review of differentiation). Find the derivatives a of the following functions.
(a) In(cos(x)) (b) arctan(e?”) (c) a® (d) x*
Hint: For parts (c) and (d), use the formula a? = el = eblna,
Problem 11.2 (Review of integration). Find the following integrals (some indefinite, some definite).
2/1 al 9 Iny
(a) f° 7+ x3 dx (b) | —_—. dr (c) | — dy
e «Vina a VY
Vin8 5
(a) | re * dx (e) [2 sin x dx (f) fe cos x dx
0
Hints: When using substitution in a definite integral you must also change the limits of integration. For (f),
you can integrate by parts twice: it will look like you are going in a circle, but it will work out!
Problem 11.3 (a first Laplace transform). Let b and s > 0 be constants. Show that:
oe b
in(bx)e ** dx = =—;.. 1
| sin(bx) e T= TD (1)
Note: The above expression is an amproper integral, because one of its bounds is oo. You solve the problem
by first computing the integral on the interval [0,4], and then taking the limit for k > oo. Le::
OO k
/ sin(bx) e ** dx = lim / sin(bx) e ** dz.

As we will see at the end of the semester, the integral (1) is called the Laplace transform of sin(bx), it is
indicated with the symbol L{sin(bx)}, and it is a function of the parameter s € R. Laplace transforms play
a central role in several fields of applied mathematics and engineering, such as differential equations, proba-
bility, signal processing, analysis of electric circuits, and control of systems. ‘To compute Ir sin(bx) e~*" dz,
you will need to apply a technique that is similar to the one used in part (f) of Problem 11.2.
Problem 11.4 (polulation growth model). In the case of infinite resources and lack of predators, the rate
of growth of a population is proportional to its own size. Therefore, indicationg with P(t) a continuous
approximation of the population size, it satisfies the differential equation P’ = aP, where a > 0 is a given
positive constant. Give the formula for the general solution of this differential equation, in terms of a and
the population size Py = P(0) at time zero. Sketch a rough graph of the general solution.
Problem 11.5. Find the general solutions to the following three separable equations:

(a) y' =a2y (b) y! = 22y? +a? —y?—1 (c) y! + y?sina =0
Problem 11.6. A radioactive substance decays at a rate proportional to the amount of the substance
present: if z(t) is the amount of substance at time t, we have x’ = —kax for some constant k > 0, where the
derivative is taken with respect to time t. If 64% of the substance remains after 10 years, what percentage
will remain after 15 years? Remark: The answer does not depend on k or the initial condition «x(0).

Problem 11.7. The second-order linear differential equations y’” = —y has the general solution given by

y(x) = cy cosx+cy sin z, where c, and cy are arbitrary constants. Such constants may be determined either

(i) by initial conditions (of the type y(0) = k; and y’(0) = ke), in which case we speak of an “initial value

problem”; or (ii) by boundary conditions (of the type y(z1) = b; and y(x2) = be, with x1; # x2), in which

case we speak of a “boundary value problem”. While initial value problems always have a unique solution,
boundary value problems sometimes do not. In fact, in order to determine the constants c,; and cy» for the
boundary conditions y(x,) = b; and y(x2) = be, we set:

C1 COSX21 + co sin x, = by or: COSZ, sinz,| |ci| — | oy

C1, CcOS%_2 + cosinx, = bo ’ cCosx2 sinz2| |co 7 by |’

which has the form Ac = b. Suppose, for simplicity, that 1; = 0. (a) Find the values of x2 for which the

matrix is nonsingular. For such values, the solution to the D.E. exists and is unique. On the other hand,

when the matrix A is singular, then the system Ac = b, and the D.E., either have no solution or infinitely
many solutions. (b) Find x2 (with rg # x1), 6; and bz for which the system Ac = b has no solutions.

(c) Find x2 (with x2 # 7), b; and bg for which the system Ac = b has infinitely many solutions.

Problem 11.8. Solve the following first order linear DEs, subject to the given conditions:

(a) y'+2y=e", y(0) =1; (b) sin(x) y’ —cos(x)y = sin(2z), y() = 0;
(c) 3y'+6ay=6e~*, y(0) =1; (d) (1+ 2?)y! + 2ry = 327, y(1) =1.

Problem 11.9 (leaky buckets of syrup). Consider two buckets of syrup, A and B. The

volume of the syrup in cans A and B is a function of time and is indicated with V,(t) ZE-

and Vp(t), respectively. At time t = 0 we have the initial conditions V4(0) = Vo (known), - CE A

and Vg(0) = 0 (the bucket B is initially empty). The buckets are stacked on top of one oe

another, like in the figure, and they both have a small hole at the bottom, of the same size: 4

the syrup in bucket A leaks into bucket B, and the syrup that leaks out of bucket B is lost. 4

(a) Assume that the rate of decrease of the volume in bucket A is proportional to the cur- \
rent volume of the syrup: V, = —kV4, where k > 0 is a known constant (that is deter- B
mined by the size of the hole). Find a formula for V,(t), that depends on Vo and k.

(b) The rate of change for the syrup in bucket B is a linear combination of the loss from the hole, and the
gain from the syrup leaked from bucket A: in other words, the differential equation Vz = —kVg—V,
holds (the second minus sign is due to the fact that V, < 0). Insert the solution for V4(t) found in
part (a) into the D.E., and solve for Vg(t) in terms of Vo and k. Remember that Vg(0) = 0.

(c) Show that limy_,.. Va(t) = 0, ie. that bucket B eventually empties. Find the time t* > 0 at which the
volume Vp(t) reaches its maximum, and find Vg(t*) (the maximum volume of the syrup in bucket B).
Express the answers in terms of Vo and k.

(d) Use Matlab to sketch the graphs of the functions V4(t) and Vg(t) when Vo = 1 and k = 1, on the same
plot. Attach the plot to your assignment. (To plot graphs with Matlab, see the back of Worksheet#10.)

Problem 11.10 (solving DEs by substitution). A differential equation of the form y’ = f(ax+by+c), with

b #0, can always be reduced to a separable equation by means of the substitution: u(x) = ax + by(x) +c.

(a) Show that, with the above substitution, by’ = u’ — a.

(b) Explain why the first order differential equation y’ = (y — 9x + 5)? is neither separable, nor linear.

(c) Use the substitution above to solve the differential equation in part (b): namely, using y’ = (y—9x+5)?,

ths substitution above (with appropriate choices of a, b, and c), and the result from part (a), find a separable

equation that only involves the function u and its derivative u’. Solve it, and substitute the function y into
. e . . ee 5 . . 1 __ 1 1 1 1
the solution. You may leave it in implicit form—you don’t have to solve for y(x). Hint: =-5 = §y=3- 6G ys3°

(d) Find an (implicit) particular solution of the equation in part (b) satisfying the initial condition y(0) = 0.