MTH5123: Differential Equations 2020
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Main Examination period 2020 – January – Semester A
MTH5123: Differential Equations
Question 1 [15 marks].
Let r be the per capita growth rate of a population in the time interval dt and N be the population density, which is the total number of individuals in this population. Note r is a constant number not a variable here.
(a) Find the general solution to the first-order ordinary differential equation (ODE),
= rN. Supposing r = 1, find the solution of this ODE when the initial
population density (t = 0) satisfies N(0) = 100.
(b) Suppose the per capita growth rate will decrease linearly with the population
density. When the population density approaches its maximum size K, the per capita growth rate decreases to 0. This yields the logistic equation,
dN N
general solution, describe how the population size changes as t → o.
Question 2 [20 marks].
(a) Find the general solution of the following ordinary differential equation
(x − 1)y\ = 2y.
ODE in (a) with the initial condition y(0) = 1 to have a unique solution in a
rectangular space D = {|x| ≤ A, |y − 1| ≤ B}. Sketch the position of this
guarantee the uniqueness of the solution in D.
(c) Use the Picard-Lindelf Theorem to show whether there exists a unique solution to the ODE in (a) with a different initial condition y(1) = 0. If not,
based on the general solution obtained in (a) and this initial condition y(1) = 0, sketch and describe all possible solutions to this initial value problem in the xy plane.
Question 3 [25 marks].
(a) Find the general solution to the homogeneous second-order linear ODE
2y\\ + y\ − 15y = 0.
(b) Use the solution in (a) to find the general solution to the inhomogeneous
second-order linear ODE
2y\\ + y\ − 15y = 6e −2x .
(c) Use the variation of parameter method to find the general solution of the inhomogeneous equation
y\\ − 5y\ + 6y = e3x cos x.
Question 4 [15 marks].
(a) Write down the general solution to the following Euler-type second order
differential equation
x2 − 4x + 6y = 0.
(b) Find the solution to the following Boundary Value Problem,
y\\ + 9y = 0, y\ (0) = 5, y() = − .
Question 5 [25 marks].
Consider a system of two nonlinear first-order ODEs,
x˙ = xy − 4, y˙ = (x − 4)(y − x).
(a) Compute all equilibria of this ODE system.
(b) Linearise the above equations around the equilibrium at y = 2 and write down the resulting linear system in matrix form. Find the corresponding eigenvalues and eigenvectors to this linearised system and write down its general solution.
stable node sink, stable spiral, saddle, unstable node source, unstable spiral)
and sketch the phase portrait for the linearised system in (b).
(d) Using the result in (b), find the solution of this linearised system in (b) corresponding to the initial conditions x(0) = 3 +^2, y(0) = 3. Determine the tangent vector to the trajectory of the solution at t = 0 and the value of x as
t → o for this specific initial condition.
2023-01-04