QUESTION

# School of Mathematical Sciences Engineering Mathematics IIA, MATHS 2201 Assignment 6 question sheet Due: Tuesday, 02/05/2017 (Week 8), by 12. It is...

Show that

(3x2 + y cos x) dx + (sin x ???? sin y) dy = 0;

is exact. Hence, or otherwise, solve the initial value problem

(3x2 + y cos x) dx + (sin x ???? sin y) dy = 0; y() =

2

:

[8 marks]

2. Verify that y1(x) = e????x is a solution of

(x + 1)y00 + xy0 ???? y = 0; x > ????1: (1)

Use reduction of order to nd another linearly independent solution and hence write

down the general solution. [12 marks]

3. Consider the problem of an undamped mechanical system upon which an external force

F acts. Suppose the external force acts only for a nite interval of time, 0  t  .

Then, the mechanical system can be described by a mathematical model in which the

displacement y(t) satises the dierential equation

y00 + 4y =

(

F(t); 0  t  ;

0; t > :

(2)

Assume that the applied force is given by

F(t) = sin 2t for 0  t  .

(a) Using the method of undetermined coecients, nd the general solution of the

nonhomogeneous dierential equation,

y00 + 4y = sin 2t for 0  t  .

(b) If the system is initially at rest, determine the values of y and y0 at t = .

(c) Write down the general solution of the homogeneous dierential equation,

y00 + 4y = 0 for t > .

(d) In this model, it is natural to insist that the solution to (2) (the displacement

y) and its rst derivative (the velocity y0) are continuous at t = . Using this

fact, and your result from part (b), determine the displacement and velocity of the

system at time t = 2.

[18 marks]