Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE (30 MARKS)
a) i) define an ordinary point and hence a singular point of a second order linear differential equation. (4 Marks) ii) Determine whether x=0 is an ordinary point of the differential equation ??" - ????' + 2?? = 0 (3 Marks) iii) Find the recurrence formular for the differential equation in ii) above. (5 Marks) b) evaluate the following integrals by expressing them as Gamma functions i. ??1 2 8 0 ??-??2???? (3 Marks) ii. ??3 8 0 ??-4?????? (3 Marks) c) Classify the following partial differential equations and find their characteristics. i. 2?????? + 4?????? + 6?????? - 2???? + ?? = 0 (3 Marks) i. 4?????? + 12?????? + 9?????? - 2???? + ?? = 0 (3 Marks) d) Transform the following partial differential equation into canonical forms i. 4?????? + 12?????? - 9?????? - 2???? = 0 (3 Marks) ii. ?????? + 2?????? + 17?????? = 0 (3 Marks)
QUESTION TWO (20 MARKS)
a) i) Determine whether x=0 or x=1 is a regular singular point of the differential equation. (1 + ??2)??" - 2????' + ??(?? + 1)?? = 0 (2 Marks) ii) Find the recurrence formulae for the power series solution around x=0 for the differential equation in i) above. (6 Marks)
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iii) show that whenever n is a positive integer, one solution near x=0 of the legendre equation. (1 + ??2)??" - 2????' + ??(?? + 1)?? = 0 is a polynomial of degree n. (6 Marks)
b) Solve the following differential equation by power series method. ??" + ?? = 0 ???????? ?? = 0. (6 Marks)
QUESTION THREE (20 MARKS) a) i) Show that ? 1 2 = ?? hence find ? -3 2 ii) Show that ?(?? + 1) = ????? ?????? ?? > 0 iii) Prove that for ?? = ?? ? Z+,?(?? + 1) = ??!. (10 Marks) b) i) Show that the Beta function is symmetric i.e ?? ??,?? = ?? ??,?? ? ??,?? ? Z+. ii) By substituting ?? = sin2 ?? in the beta function, show that ?? ???? = 2 sin2??-1 ??cos2??-1 ?????? ?? 2 0 . (10 Marks)
QUESTION FOUR (20 MARKS)
a) find the general solution to a Bessel equation of order zero given by ??2??" + ????' + ??2?? = 0 (10 Marks) b) i) prove that (i) ????-1 ?? - ????+1 ?? = 2????(??) (ii) ?? ???? ????+1????+1(??) = ????+1????(??) (10 Marks)
QUESTION FIVE (20 MARKS)
a) A stretched string of length 20cm is set, oscillating by replacing its midpoint a distance 1cm from its rest position and releasing it with zero initial velocity. Solve the wave equation ??2?? ????2 = 1 ??2 ??2?? ????2 where ??2 = 1 and determine the resulting motion ??(??,??).(12 Marks)
b) A bar of length 2m is fully insulated along its sides. It has initial uniform temperature of 10°?? and at t=0, its ends are plugged into an ice bucket and maintained at a temperature 0°??. Determine an expression for the temperature at a point p at a distance x from one end at any subsequent time t. (8 Marks)
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