(1) A cold beer with an initial temperature of

1

35 F

warms up to

40 F

in

10 minutes while sitting in a room of temperature

70 F

.

(i) What will be the temperature of the coke after t mins? Please

show details of calculations. [8 marks]

(ii) What is the temperature of the coke after 20 minutes?

[2 marks]

(2) Consider the following second-order differential equation

2

2

dx

yd

dy

x

1

2/32

2

xxyx

dx

x cos

4

(i) Given that one of the solution is

y

sin

1

x

x

. Find the other

solution using method of reducing the order and thus write

down the complementary solution (

y

). [10 marks]

(ii) Find the Wronskian

),(

yyW

of the two solutions (i) and

decide whether they are linearly independent or dependent.

11

c

[2 marks]

(iii) Find the particular solution (

y

) in its simplest form by

solving the non-homogeneous part of the equation using

p

method of variation of parameter. Then write down the

general solution. [8 marks]

(3) Use the method of undetermined coefficient to find the general

solution of the given differential equation.

)4(

cos520’4”5”’124

x

x

[10 marks]

2

2

exeyyyyy