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Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2010
QUESTION ONE – (30 MARKS)
(a) Define the Wroskian of ?? real function ??1,??2 …????. (2 Marks)
(b) Find the Wroskian of the functions x f 2 sin1 ? x f 2 cos2 ? (3 Marks)
(c) Given the differential equation
0442
2 2 ? ?? y dx dy x dx yd x
(i) Prove that ?? = ?? is a solution of the equation, hence find a linearly independent solution of the equation by reducing its order. (7 Marks)
(ii) Write the general solution of the equation. (2 Marks)
(d) Prove that the equation below is solvable. ???? ???? + ??2?? - ???? ???? + ??2?? - ???? ???? = 0 (4 Marks)
2
(e) Find a first integral of
22
2
2
xy
dx dy
yx
dx yd
dx dy ? ? (2 Marks)
(f) Locate and classify the singular points of the equation ? ? ? ? 0 122 2 2 2 234 ? ????? y x dx dy x dx yd xxx . (3 Marks)
(g) Prove that the Legendre polynomial of order three is given by ? ? x x x xp 2 35 3 3 ? ? (3 Marks)
(h) Prove that the equation
xy
dx dy
x
dx yd
x
dx yd sin532 2 2 2 3 3 ???? ? ? ? ? 2 7 4,34 '''' ? ?? yy has a unique solution which is defined for all ??. (4 Marks)
QUESTION TWO – (20 MARKS)
(a) Show that ??, ??2 and ??4 on solutions of 0 884 2 2 2 3 3 3 ? ??? y dx dy x dx yd x dx yd x , given that ??? x0 (6 Marks)
(b) Can you say that ?? = ??1?? + ??2??2 + ??3??4 is the general solution of the above differential equation? Give reasons for your answer. (4 Marks)
(c) Prove that the functions 1,??,??2 are linearly independent. (2 Marks)
(d) Form the differential equation whose roots are 1, ??,??2 . (5 Marks)
(e) Prove that ??-?? ?? = -1 ?? ???? ?? . (3 Marks)
QUESTION THREE – (20 MARKS)
(a) Show that ?? = ?? is a solution of the equation. (4 Marks) ? ? ? ? ? ? 0 cos2sin6cos2sin6cossin3sin 2 2 2 3 3 3 2 ? ?????? y xxx dx dy xxxx dx yd xxxx dx yd xx s
3
(b) Reduce the third order equation above to a second order equation and solve it. (6 Marks) (c) Consider the equation ?? = ?? + ?? ???? + ???? ?? + ?? ???? + ???? ?? + ?? ???? = 0 show that the condition of integrability is satisfied hence solve the problem. (10 Marks)
QUESTION FOUR – (20 MARKS)
(a) Show that ?? = 0 is an ordinary point for the equation ? ? 0 31 2 2 2 ? ??? xy dx dy x dx yd x (2 Marks)
(b) Find the power series solution for the equation given in (i) above. (7 Marks)
(c) State Rodrigue’s formula for the nth order polynomial ???? ?? . (1 Marks)
(d) Find ???? ?? and ???? ?? using Rodrigue’s formulae. (3 Marks)
(e) Prove that Bessel’s function of order zero is given by
???? ?? = 1 - ??2 22
+
??4 22.42 -
??6 22.42.62
+ ? (3 Marks)
(f) Prove that there exists a unique solution for the equation
xeyx
dx dy
x
dx yd
23 3
2
2
??? ?? ?? 5 1,1 '' ? ?? yzy (4 Marks)
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