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