School of Mathematics and Statistics University of New South Wales
MATH2221
Higher Theory and Applications of Di erential Equations
SESSION 2, 2018 ASSIGNMENT 1 GROUP A
Q1 Consider the following ODEs
du
dx =juj,x2R and
dv
dt = v
1=2 ,t2[0;1]:
a). Show that these permit solutions of the form. u = Aex and v = Bt2 respectively.
b). Solve these for the cases u(0) = 0 and v(0) = 0.
c). Is the solution unique in either case? Explain your answer.
Q2 Consider the function
f(x) = 1 x for 0 1=2)?
Explain your answer.
Q3 You are working in collaboration with glaciologists who are storing ice cores. The cores are
long and thin and perfectly insulated save a small amount of heating at a rate at one end. The
glaciologist hope to balance this warming with cooling at a rate at the other end. You have
determined that the ice core obeys the following boundary value problem
ut uxx = 0; ux(0) = ; ux(l) =
where u is temperature, t is time and l is the length of the core.
a). What should be such that the ice core’s temperature remains stable (ut = 0)?
b). Assuming = 1, l = 10 and the average temperature of the core is -15, what is the solution
for u in the stable case?
b). If the cooling mechanism were to fail ( = 0) how long would it take before the ice core started
to melt (i.e. when would u rise above 0 at any point)?
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School of Mathematics and Statistics University of New South Wales
MATH2221
Higher Theory and Applications of Di erential Equations
SESSION 2, 2018 ASSIGNMENT 1 GROUP B
Q4 The homogeneous Van der Pol equation is given by
x+ (x2 1) _x+x = 0 with > 0:
a). Write down a pair of rst-order ODEs that is equivalent to the Van der Pol equation and nd
the equilibrium point.
b). It can be demonstrated (although you are not required to do so) that for the equilibrium point
a there exists a positive real number such that for any jjx(0) ajj there exists a positive real
number M such that jjx(t) ajjM. Does this imply that a is stable?
c). Linearise the system and determine whether the equilibrium point is linearly stable.
d). For what value of is K =jjxjj a rst integral of the system of equations.
Q5 You are working in collaboration with biologists who are growing algae in a lab. You have
deciphered that the algae generally obeys the following ODE
t2u00 +tpu0 +qu = f(t):
The biologists do not set initial conditions. Rather, once the algae is growing, they measure u at
at t = 1 and t = 1 and use this to predict (empirically) what the result will be at t = 2. In their
rst set-up with p = 2 and q = 4 they were able to consistently predict u. In a second set-up
where p = 7 and q = 8 they were not able to consistently predict u.
a). Explain the behaviour of the rst set-up.
b). Explain the behaviour of the second set-up.
b). What advice would you give to the biologists with regard to their approach to measuring and
predicting u?
Q6 Consider the parameterised Bessel equation
z2 d
2u
dz2 +z
du
dz + (k
2z2 2)u = 0 (1)
where k and are real numbers.
a). Convert (1) into Sturm-Liouville form.
b). What equation must k satisfy such that (1) and the boundary conditions: u0(0) = 1 and
u(l) = 0 are satis ed.
c). Graphically show the rst 3 corresponding eigenfunctions. Correctly label your graph and give
some details of how it was made.