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Unformatted text preview: AMATH 250  Final Exam, Winter Term 2007 Page 1 of 11 P 1. Find the general solution to the following DES: {21 a) =
[6] b) = gm — 1r
[6] c) g = —2xy + 2x. [10] 2. Solve the following initial value problem (IVP) for { y”——4y’+ 3y = 6‘“ +sinm = 07 yl(0) = 1 [2] 3. a) The diffusion of heat in a slab of material is governed by the partial differential equation (the heat equation)
3T 62T _=K_ 6t 63:2 ’ where the temperature T varies only in the m—direction. The constant K: is called the thermal dzﬂusivity, and depends on the nature of the material. Find the dimensions
of FL. ’ [10] b) A little cooking problem how does the cooking time of a roast depend on the size
of the roast? The physical quantities are the cooking time At, the volume of the roast V, the
thermal diffusivity K. of the roast (see a)), the initial temperature difference ACE :
T, — To and the ﬁnal temperature difference ATf = Tf — To. Here To is the oven
temperature, and Ti, Tf denote the initial and ﬁnal temperature of the centre of the
roast, respectively. Show that At is proportional to V2/3. [HINT: Instead of the usual
M, L, T, the three fundamental quanitities in this problem are 6, L, and T where 0
represents temperature] 4. The DE dN N
E=T(1_E)N_h‘ where r, K and h are positive constants, describes a population of ﬁsh with natural growth
rate coefﬁcient 7“, carrying capacity K and a constant harvest rate It. 1
[6] a) Show that if h < ErK there are two equilibrium solutions N (t) = N1 and N (t) = N2
where N1 and N2 are constants with 0 < N1 < N2. Find N1 and N2.
[7] b) Give a qualitative sketch of the solution curves in the case h < irK. Discuss the
long term behaviour of N(t) in the two cases N(O) > N1 and (ii) N(O) < N1.
rs [HINT: You do NOT need to solve the DE], [4] 5. a) Use the deﬁnition of the Laplace transform to evaluate £{tea‘}, and state the values
of s for which it is valid. You may use the formula sheet for parts b) and c). 1 3
[2] b) Evaluate [I {——————~s2 + 28 + 10}. AMATH 250  Final Exam. Winter Term 2007 Page 2 of 11 [9] c) Solve the NP
y’+y = 1—H(t—a)
{ 31(0) = yo
where a and yo are positive constants and H is the Heaviside step function. Sketch
the solution y(t) for t 2 0. 6. Consider the coupled mixing tanks shown, where f is a ﬂow rate (volume/ time) and c is
a nonzero concentration (mass/time). Let m1(t) and m2(t) be the mass of salt in tank
#1 and 2, respectively, and suppose that the volume of brine (salt—water mixture) in each tank is V, a constant.
C"/ [10] a) Derive two DE’s describing this situation, and show that they may be expressed as the vector DE 2
_./ _ _ 1 _. 1
a: —(2 _2)$+(0) after non—dimensionalizing.
[10] b) If c was zero, the vector DE would have been a,_ —2 1 s
._(2 _2)._ Find the general solution to this homogeneous DE, and sketch the phase orbits, in
the quadrant(s) of the plane which have physical meaning. [5] c) Find the general solution to the inhomogeneous DE using the method of variation
of parameters. (You may use the Laplace transform method for partial credit and/ or to check your answer). 7. Consider a sinusoidally forced oscillator, whose displacement y(t) is described by
my” + cy’ + Icy = f0 cos(wt) (1) where m=mass, c=damping constant, k=spring constant, f0: amplitude of applied force,
and w=forcing frequency. All of these constants are positive. The DE (1) can be written in dimensionless form
Y” + 2(Y’ + Y = cos QT (2) where ’ denotes differentiation with respect to dimensionless time 7' and Y(7’) is the di—
mensionless displacement. (Thus, only two parameters appear instead of ﬁve). [5] a) Write the DE (2) as a ﬁrst—order vector DE. [5] b) Suppose that f0 = 0, so that the cos QT does not appear. On an assignment, you
saw an example of an underdamped oscillator starting from rest, and showed that it
reached maximum speed just before it reached the equilibrium position (y(t) = 0)
for the ﬁrst time. The question is: In general, with f0 = 0 and arbitrary initial
conditions, is the speed of the oscillator increasing, decreasing, or constant each
time it passes through the equilibrium position? Does it depend on whether the oscil—
lator is overdamped, underdamped, or critically damped? Use phase orbits to explain. ...
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 Winter '07
 joseph
 Math

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