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Math 327: Methods Of Applied Mathematics 1 Question Paper

Math 327: Methods Of Applied Mathematics 1

Course:Bachelor Of Science (General) & Bachelor Of Education (Science)/ (Arts)

Institution: Chuka University question papers

Exam Year:2013

CHUKA

UNIVERSITY

UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATIONS FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE (GENERAL) & BACHELOR OF EDUCATION (SCIENCE)/ (ARTS)
MATH 327: METHODS OF APPLIED MATHEMATICS 1
STREAMS: BSC (GEN), BED (SCI & ARTS) Y3S2 TIME: 2 HOURS
DAY/DATE: THURSDAY 25/4/2013 8.30 AM – 10.30 AM
INSTRUCTIONS:

Answer Question ONE (Compulsory) and any other TWO Questions
Adhere to the instructions on the answer booklet
Do not write on the Question paper.

QUESTION ONE (COMPULSORY) (30 MARKS)

(a) Using power series method solve the differential equation

?2y?^''-2y=0 [5 Marks]

(b) Show that

J_4 (x)=(48/x^3 -8/x) J_1 (x)+(1-24/x^2 ) J_0 (x) [6 Marks]

(c) Express f(x)=?4x?^3+?6x?^2+7x+2 in terms of Legendre polynomials. [6 Marks]

(d) Find the Fourier series representing f(x)=x, 0<x<2p [5 Marks]

(e) Use the method of separation of variables to solve the differential equation

(d^2 u)/?dx?^2 =du/dt [6 Marks]

(f) Define the term periodic functions [2 Marks]

QUESTION 2 (20 MARKS)

(a) Find the Laplace transform of sin?2t/t [4 Marks]

(b) Find the inverse Laplace transform of (s^2+3)/(s(s+9)) [6 Marks]

(c) Use Laplace transforms to solve the simultaneous differential equations below

dx/dt+y=0 and dy/dt-x=0 under the conditions x(0)=1, y(0)=0
[6 Marks]

(d) Obtain a Fourier expression for f(x)=x^3,for-p<x<p [4 Marks]

QUESTION 3 (20 MARKS)

(a) Given that f(x)=x+x^2 for -p<x<p, find the Fourier expansion of f(x)
[9 Marks]

(b) Prove that ?4J?_n^'''' (x)= J_(n-2) (x)+J_(n-2) (x) [5 Marks]

(c) (i) Solve the Bessels differential equation x^2 y^''''+?xy?^''+x^2 y=0, given that

? J?_n (x)= [3 Marks]

(ii) From (i) above obtain the series for J_1 (x), hence find J_1 (0) [3 Marks]

QUESTION 4 (20 MARKS)

(a) Obtain the solution of the wave equation (d^2 y)/?dt?^2 =(c^2 d^2 y)/?dx?^2 using the method of separation of
variables. [14 Marks]

(b) Given that the differential equation x (d^2 y)/(dx^2 )+dy/dx-y=0 has a solution in the form
obtain the indicial equation and find its roots. [6 Marks]

QUESITON 5 (20 MARKS)

(a) Solve the equation

(1+x^2 ) (d^2 y)/?dx?^2 +x dy/dx-y=0 given that its series solution is given by
[12 Marks]

(b) Using Laplace transforms, find the solutions of the initial value problem

y^''''+9y=9u(t-3) given that y(0)= y^'' (0)=0

Where u(t-3) is the unit step function. [8 Marks]

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Math 327: Methods Of Applied Mathematics 1 Question Paper

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