Reference no: EM132248562
Questions -
Q1. Consider the initial value problem
(x)dy/dx = y - 2, y(a) = b
Show that it
(i) Has a unique solution if a ≠ 0.
(ii) Has infinitely many solutions if a = 0 and b = 2.
(iii) Has no solution if a = b = 0.
Q2. Suppose that the population y of a certain species of fish in a given area of the ocean is described by the logistic equation
dy/dt = ay(1-by) - 1/8
Where a, b are positive constants and y > 0 is a function of t.
(i) If a = m/n and b = 1/n, with m and n being the largest two distinct digits in your ID number and m < n. Then use MAPLE and sketch a direction field for your differential equation and include a sufficient number of solution curves and include the graph into your answer sheet.
(ii) If a = m/n and b = 1/n, with m and n being the largest two distinct digits in your ID number and m < n. Then use Maple and find the equilibrium solutions and determine whether they are asymptotically stable or unstable.
(iii) ½ > a/b, If show that y decreases as t increases regardless of the value of the initial.
(iv) If ½ = a/b find a single equilibrium point and show that this is a semistable point.
Q3. Consider the following equation
dy/dt = (m-6)y - y2, y(0) = y0
Where y is a function of t.
(i) Define stable, semistable, unstable equilibrium (critical) points and Bifurcation point.
(ii) If m being the largest digit in your ID number, find all the equilibrium points for the equation and determine whether each critical point is asymptotically stable, semistable or unstable.
(iii) For what value of m is there a bifurcation point? Briefly explained why the value obtained is a bifurcation point.
(iv) Solve the above equation subject to initial condition ??(0) = 1 and m being the largest digit in your ID number.
Q4. Consider the following system
x'1 = x2
x'2 = -x1 + x2 (1)
(i) Write the above equation in the form of a matrix equation
X'(t) = BX(t) (2)
where B is a 2 x 2 matrix whose entries are real numbers.
(ii) Use Matrix method to find a general solution of the above system.
(iii) Is the critical point of this system stable? Provide a brief explanation.
Note - It is ESSENTIAL to show all working in your solution to question 1.