The a-s-s-e-s-s-m-e-n-t will be 7 calculation questions. Please see the questions shown in the screenshot. I will send you all info after being hired, eg PPTs, student access etc. Please send a draft in 12hrs -1 day time, day 2, and day 3 as well. + Will need to draft some questions to ask the teacher and revise base on feedback (Send bk ard in 1 day max)

MATH7000 PROBLEM SET

SEMESTER 2 2022

Due at 4:00pm 19 October. Marks for each question are shown. Total marks: 30

Submit online via the MATH7000 Problem Set submission link in Blackboard.

(1) (5 marks.) Consider the non-homogeneous ODE

√

d2 y

dy

−

y

=

1 + x2 .

+

x

dx2

dx

The corresponding homogeneous ODE is

(1 + x2 )

d2 y

dy

− y = 0.

+

x

dx2

dx

(i) Show that by the variable change x = sinh t, the homogeneous ODE is transformed to

d2 y

− y = 0.

dt2

(ii) A fundamental set of solutions of this equation can be chosen to be y1 =

sinh t and y2 = cosh t. Deduce that a fundamental

set of solutions of the

√

homogeneous ODE are y1 = x and y2 = 1 + x2 .

(iii) Find the general solution of the non-homogeneous ODE.

(1 + x2 )

(2) (5 marks.) Determine for what real numbers a and b, the map f : R2 × R2 given

by

f ((x, y), (x0 , y 0 )) = axx0 + byy 0

is an inner product.

x1

y1

(3) (5 marks.) Suppose that the orthogonal transformation

= P

x2

y2

2

2

2

2

transforms f = x1 − 4×1 x2 + 4×2 into f = 4y1 + 4y1 y2 + y2 . Find the orthogonal

matrix P .

(4) (4 marks.) Suppose that f (x, y) is continuous on D = {(x, y)|x2 + y 2 ≤ y, x ≥ 0}

and satisfies

p

8

f (x, y) = 1 − x2 − y 2 −

f (x, y) dxdy.

π

x

D

Find f (x, y).

y

y

(5) (4 marks.) You are given that

V

f (x, y, z) dxdydz =

f (x(u, v, w), y(u, v, w), z(u, v, w)) |J| dudvdw,

R

1

under the variable transformation x = x(u, v, w), y = y(u, v, w) and z = z(u, v, w),

which maps the region R (in the uvw-space) one-to-one into the region V (in the

xyz-space). Here J is the Jacobian of the variable transformation defined by

∂x ∂x ∂x

∂u

∂v

∂w

∂(x, y, z)

∂y

∂y

∂y

.

J=

= det ∂u

∂v

∂w

∂(u, v, w)

∂z

∂z

∂z

∂u

∂v

∂w

Use an appropriate variable transformation to evaluate the triple integral

s

2

3/2

x

y2 z2

1−

dV,

+ 2 + 2

a2

b

c

y

V

2

2

2

where V is the region enclosed by xa2 + yb2 + zc2 = 1.

(6) (3 marks.) Suppose that z = f (x, y) is continuous and differentiable and satisfies

the relation

∂z

∂z

2

2

+y

= e(x−1) +y .

(x − 1)

∂x

∂y

Evaluate

Z

∂z

∂z

dy −

dx,

∂y

c ∂x

√

where C is the curve from (2, 0) to (0, 0) along y = 2x − x2 .

R

R

Hint: Note that c F(r) · dr = c F(x, y) · T(x, y) dS, where T(x, y) is a unit

tangent vector to C at point (x, y) on C.

(7) (4 marks.) Compute the line integral

I

(4x − y)dx + (x + y)dy

,

4×2 + y 2

C

where C is the circle x2 + y 2 = 2 traversed in an anti-clockwise direction.

2