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M2A1 Exam 2007
1. Consider the pair of ordinary differential equations
x ˙ = −x − xy , y˙ = 1 − x − y ,
where the dot refers to differentiation with respect to time t.
(i) Show that they have two fixed points, one of which lies on the y-axis and the other lies in the lower half-plane. (ii) Find the local eigenvalues and eigenvectors of the Jacobian matrices corresponding to each point. Hence show that these points are classified as a stable node and a saddle respectively. (iii) Sketch the phase plane showing how phase trajectories connect between these two points. (iv) In particular sketch the trajectory in the upper half-plane that is is tangent to the x-axis at the point (1, 0). Use the differential equations to show that near this point, this trajectory can be approximated by
y = 12 (x − 1)^2.
2. Find the critical points of the nonlinear system of ordinary differential equations
x ˙ = y , y˙ = y + 1 − x^2.
Classify these critical points and sketch a selection of orbits in the phase plane. On your diagram, you should show the locus of points on which dy/dx = 0, the locus of points on which dy/dx = ∞, and the signs of dy/dx in each sector into which the phase plane is divided by these loci.
3. Use the disturbance function method to show that the integral
I =
∫ (^) x 2
x 1
f (x, y, y′) dx
takes stationary values if y(x) satisfies the Euler-Lagrange equation
fy −
d dx
fy′^ = 0.
y(x 1 ) and y(x 2 ) take fixed values and subscripts denote partial derivatives. You may assume that f has continuous 2nd partial derivatives.
(i) Write down an expression for the total derivative d/dx. (ii) Show that the Euler-Lagrange equation can be written in the form
fx +
d dx
(y′fy′^ − f ) = 0.
Hence show that if y(0) = 0 and y(1) = 1 and
I =
0
y′^2 + β^2 y^2
dx ,
then I takes stationary values when y(x) satisfies
y =
sinh βx sinh β
4. The positive part of the y-axis is the principal axis of a cone whose vertex lies at the origin.
The curved surface of the cone is represented by x^2 + z^2 = y^2 , which is parametrized by x = y cos θ and z = y sin θ. Show that the arc length of a curve on this surface, with end points represented by θ 1 and θ 2 , is given by
arc length =
∫ (^) θ 2
θ 1
ds =
∫ (^) θ 2
θ 1
dy dθ
dθ.
Show that the arc length takes stationary values when y and θ satisfy the differential equation
2 c^2
dy dθ
= y^2
y^2 − c^2
, c = const.
Show that this differential equation is satisfied by
y = ±c cosec
θ + δ √ 2
, δ = const.
You may assume the Euler-Lagrange equations in the form (y′^ = dy/dθ)
fθ +
d dθ
(y′fy′^ − f ) = 0.
5. Traffic flows on a road with a density ρ(x, t) according to the kinematic wave equation
ρt + (1 − ρ)ρx = 0. (a) Show that for initial data ρ(x, 0) = f (x) there is an implicit solution of the form ρ = f (x − (1 − ρ)t). (b) Consider the two sets of initial conditions: (i)
f (x) =
x 0 ≤ x ≤ 1 , 2 − x 1 ≤ x ≤ 2 , 0 otherwise. (ii)
f (x) =
1 x ≤ 0 , 1 1+x x^ ≥^0. Show that a shock develops from (i) but not from (ii). Does this shock develop on the backwards-looking face or the forwards-looking face? (c) For case (i) in Part (b) above, find the solution ρ(x, t) explicitly in terms of x and t. (d) Show that for this version of the kinematic wave equation, the propagation velocity is less than the flow velocity.
