Question 1:

(a) Find all roots of equation z³ + (1 + j)z² + (6 + j)z + 6 = 0 and show them in the Argand diagram.

(b) Find the real and imaginary parts of the complex number

  Z = 5j j-5/2+j + 4e^-jΠ/2 

Present this number in the rectangular, polar and exponential forms. Show it in the Argand diagram.

Question 2:

Find and plot the Cartesian equations of the loci of z given by the following equations:

a) |z + 3/12j - 4z|

b) |z - 3j| + |z + 7j| = 12

c) arg (z + 2 - 3j) = Π/3

Question 3:

Determine which of the following differential equations are separable. In each case explain why the differential equation is or is not separable. If separable find the general solution to the differential equation.

Hint: Your solution may be need to be left in quadrature, i.e. as an im­plicit function.

a) y' = exp (3 ln x - In tan y),

b) y' = (tanx + tany)cosxcosy,

c) y' = exp(1/2ln y+ ln x + 2ln 2).

Question 4:

Find the general solution of the differential equation

y' = 2x + cotan x / 3_y²

Determine the integration constant using the initial condition y(r/2) = 0. Present the particular solution subject to this initial condition in the explicit form.

Question 5:

Determine whether the differential equation

dy/dt +6yt = 12t.

- is linear or nonlinear?

separable or non-separable?

Find an asymptotic (terminal) solution (i.e. the solution at t → ∞). Sketch a particular solution subject to the initial condition y(0) = 4.

Question 6:

Consider the initial value problem

y. = cos y + 3/y,  y(0) = 0.5.

Use Euler's method with 5 steps to estimate y(0.1) up to three decimal points. Sketch accurately the direction field in the domain 0 ≤ t ≤ 0.1, 0.2 ≤ y ≤ 1.4. Determine whether the approximate solution overestimates or underestimates the exact solution.

Question 7:

Identify which class the following differential equation belongs to? (Sepa­rable or non-separable, linear or nonlinear; if linear, homogeneous or non-homogeneous)?

y. = y + 3/t² + 5/ t

Write it in the standard form and find a solution subject to the initial con­dition y(1) = 2.

Question 8:

The growth rate of bacterial populations in time is proportional to their concentration n(t) with the coefficient of proportionality a.

(a) Write down this statement as a differential equation.

(b) Solve the differential equation in the previous item in the general form for the bacterial populations, that is derive the formula for the solution subject to the initial condition n(0) = n_o assuming that a and n_o are given parameters;

(c) Find at what time the concentration of bacteria becomes double its initial value.