Question 1: [10 marks]
a) Find all roots of equation z
2
– 4 z + 13 = 0 and show them in the Argand
diagram. [3 marks]
b) Find the real and imaginary parts of the complex number [7 marks]
3j 4 5j+2 2 10j .
2j+1
z e
? ?
? ? ?
Present this number in the rectangular, polar and exponential forms.
Show the number obtained in the Arg
2
Hint: Solution may be not integrable in terms of elementary functions. In this case it can be left
in quadrature, i.e. in terms of integral. Don’t give up in advance and make sure that this is the
case.
a)
2 2
y x y
?
? ?1 cos cos
; [5 marks]
b)
y x y
? ? ? ln? ?
; [5 marks]
c)
4ln ? ?
y
x
y e
? ?
?
?
? ? ? ?
[10 marks]
Question 4: [15 marks]
Find the general solution of the differential equation:
2
6 cos
x
y
y y
?
?
?
. Determine the integration
constant using the initial condition y(–2) = 0. Present the particular solution subject to this initial
condition in the explicit form (think in which form the explicit solution can be presented y(x) or
x(y)?).
Question 5: [15 marks]
Determine whether the differential equation
x
xy y e
?? ?
- is linear or nonlinear?
- separable or non-separable?
- homogeneous or non-homogeneous?
Find a particular solution subject to the initial condition y(1) = 2e. Plot the solution in the
interval x ? [0.25, 4].
Question 6: [20 marks]
The water temperature in a tank takes 10 minutes to decrease from 100°C to 60°C. The
temperature of the surrounding air is kept constant at T0 = 20?C.
a) Assuming that the rate of temperature decrease is proportional to the difference of current
water temperature and constant air temperature, find the differential equation describing
water temperature as a function of time.
b) Solve the differential equation and find at what time water temperature attains 30?C?
3
Question 7: [25 marks]
a) Determine the type of the differential equation
y yt t ? ? 6 12
(is it linear/nonlinear,
separable/non-separable, homogeneous/non-homogeneous)?
b) Find two particular solutions subject to two initial conditions y(0) = 4 and y(0) = 0.
c) Plot the solutions for t > 0 and find their asymptotics when t ? ?.
Question 8: [20 marks]
a) Determine the type of the differential equation
3
cos cos sin dy
x x y x
dx
? ?
(is it linear/
nonlinear, separable/non-separable, homogeneous/non-homogeneous)?
b) Find the particular solution subject to the initial condition y(? /4) = 11/6.