Department of Mathematics and Philosophy of Engineering

MPZ4230-Engineering Mathematics II

Assignments NO 1

Due Date: 09 -11 -2014 Academic Year 2014/2015

1. (a) Find the values of the constants A and B so that y = ex(A sin 2x + Bcos2x),

satisfies the differential equation:d4y

dx4+ 2y = exsin2x

(b) Show that∂2z

∂x2− 2

∂2z

∂x∂y+∂2z

∂y2= 0; where z = xf(x+ y) + yg(x+ y)

(c) Prove that if (x, y) =1√ye

−(x−a)2

4y ; then fxy(x, y) = fyx(x, y).

2. The work that must be done to propel a ship of displacement D for a distance S in

time t is proportional to kS2D

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t2. Find approximately the percentage increase of work

necessary when the displacement is increased by 3%, the time diminished by 2%, and

the distance diminished by 4%.

3. (a) Find the dimensions of the rectangular box open at the top of maximum capacity

whose surface is 432 sq.cm.

(b) Show that the height of the right circular cylinder of maximum volume that can

be inscribed in a given right circular cone of height h is1

3h.

4. (a) Given that the function f(x, y) = 2x+ 4x2 − y2 + 2xy2 − x4 − y4.

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i. Write down all the first and second order partial derivatives of the function

f(x, y).

ii. Find all the maximum, minimum and saddle points of the function f(x, y).

(b) f(x, y, z) = xe2yz;P (3, 0, 2);u = (2

3,−2

3,1

3)

i. Find the gradient of f .

ii. Evaluate the gradient at the point P .

iii. Find the rate of change of f at P in the direction of the vector u.

5. (a) State the Cauchy Riemann equation for the function f(z) = u+ iv to be analytic.

(b) Show that the function u(x, y) = sin x cosh y is a harmonic function.

(c) Find the harmonic conjugate v(x, y).

(d) Express f(z) = u+ iv as a function of z = x+ iv.

6. (a) Using the Cauchy Integral Theorem, evaluate the contour integral

∮c

5zdz

(z + 1)(z − 2)(z + 4i)Where C is

i. the circle|z| = 3;

ii. the circle|z| = 5

(b) Compute the following integrals.

i. I =

∫ 2π

0

dθ

(5− 3 sin θ)2

ii. I =

∫ π

−π

dθ

1 + sin2 θ

• Please send the answer script of your assignment to the following address and note

that your assignment must be REACHED to the department on or before the due

date scheduled in the activity diary

Coordinator - MPZ4230,

Department of Mathematics and Philosophy of Engineering,

Faculty of Engineering Technology,

2

The Open University of Sri Lanka,

Nawala, Nugegoda.

Nugegoda.

End

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