Population model

In some populations, the amount of births is directly proportional to the population at any given point in time and the amount of deaths is directly proportional to the square of the population at any given point in time. 1. Write an equation that models the change in a population that fits the above description. Make sure t ...continues

Differentiation : Radius of Convergence for Power Series

Consider the differnetial equation y'(x) + xy(x) = 0 with y(0) = 0 Look for a solution of this problem of the form y(x) = A + B + Ce^-x + De^-1/2x^2 Use the fact that y must satisfy the equation and the initial conditions to identify the constants A,B,C and D. By setting u = -x^2/2 in the power series for f(u) = exp{u}, ...continues

Mathematical Modelling

Please see the attached file for the fully formatted problems. A long, thin rod is clamped at its lower end and a mass M is attached to its upper end. The coordinates (x,y) of any point on it satisfy the equation. where E, I and a are constants. Given that x = 0 when y = 0 and x = a when y = L show that where H ...continues

Simulation : Skydiver in Free-fall

A skydiver, weighing 70kg, jumps from an aeroplane at an altitude of 700 metres and falls for (T) seconds before pulling the rip cord of his parachute. A landing is said to gentle if the velocity on impact is no more than the impact velocity of an object dropped from a height of 6 metres. The distance that the skydiver falls d ...continues

Scale model

I have a model of Mercury that is 5 inches in diameter the actual diameter is 4880 km. What is my scale?

Product rule and chain rule

I have done most of the calculations but require confirmation.

Lorenz Equations and Equilibrium

Please see the attached file for the fully formatted problems. Show that for , the equilibrium (x*, y*, z*) = (0, 0, 0) is globally (nonlinearly) stable for the Lorenz system. That is, any (x(t), y(t), z(t)) would eventually approach (0, 0, 0) as . Consider the “volume” (a) Show that, using the Lorenz equations ...continues

Graphing, Limits, Continuity, Countability, Surjectiveness, differentiability and Permutations

This is a test on Sets, Functions and Permutations. There are several questions. Please see the attached file for the fully formatted problems.

Past exam, question B6 - Mathematical Modelling

I would be grateful if someone could give me the solutions and workings for Q B6 of the attatched paper.

Mathematical Modelling : Elliptic Partial Differential Equation

Please see the attached file for the fully formatted problem. Show that the PDE ... is of elliptic type for 01and y<0 (and all x does not equal 0)?