 |
PDE801S - PARTIAL DIFFERENTIAL EQUATIONS - 1ST OPP - JUNE 2023 |
|
|
1 Page 1 |
▲back to top |
nAmlBIA unlVERSITY
OF SCIEn CE Ano TECHn OLOGY
FACULTYOF HEALTH,NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS, STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: BACHELOR OF SCIENCE HONOURS IN APPLIED MATHEMATICS
QUALIFICATION CODE: 08BSHM
.LEVEL: 8
COURSE CODE: PDE801S
COURSE NAME: PARTIAL DIFFERENTIAL EQUATIONS
SESSION: JUNE 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 98
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Prof A.S Eegunjobi
MODERATOR:
Prof Sandi le Motsa
INSTRUCTIONS
1. Answer ALL the questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Including this front page)
|
|
2 Page 2 |
▲back to top |
PDE 801S
PARTIAL DIFFERENTIAL EQUATIONS
JUNE 2023
QUESTION 1 [24 marks]
l. Form partial differential equations
(a) by eliminating the arbitrary functions ¢ from the relation
xyu(x, y) = ¢(x + y + u(x, y))
(7)
(b) by eliminating the arbitrary functions f and g from the relation
u(x, y) = f(x + ay) + g(x - ay)
(7)
(c) by eliminating the arbitrary constants a, b and c from the equation
x2 y2 z2
2a + b2 + 2C = l
(10)
QUESTION 2 [24 marks]
2. Solve the following first order PDE
(a) DDxu+ 3DDuy = 5u + tan(y - 3x) ·
(8)
(b) u%-~ut;= (x+ y)2 + z2
(8)
(c) x(y - z)i; + y(z - x)i~ = z(x -y)
(8)
QUESTION 3 [20 marks]
3. (a) Reduce to normal form and hence solve
az az (y - 1)-8f)2xz2 - (y2 - 1)-a-8x2fJzy + y(y - 1)-88y2z2 + -ax - -f)y = 2ye2x( 1 - y )3
provided y -=I=1
(10)
(b) Reduce to normal form
Zxx + 2zxy + 5zyy + Zx - 2zy - 3z = 0
(10)
QUESTION 4 [30 marks]
4. (a) Determine the displacement y(x, t) for a taut string with fixed endpoints at :r: = 0
't) and x = l, initially held in position y = y0 sin3 ( 1 and released from rest.
(15)
(b) Find the solution of the Cauchy problem
x E IR, t > 0, u(x, 0) = f(x), Ut(X, 0) = g(x), x E IR.
(15)
End of Exam!