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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:2013

QUESTION ONE (30 MARKS)
a) Distinguish between each of the following terms; i. Ordinary differential equation and partial differential equation. (2 Marks) ii. Consider the differential equation ??2?? ????2 3 2 + 5???? ???? + 3?? = 0. State its order, degree and state if it’s linear or non-linear. (3 Marks) b) Show that ?? = ??2 is a solution of the equation 3??2?? ????2 + 5 ???? ???? = 10?? + 6. (4 Marks) c) Find the differential equation whose solution is ?? = ??sin3?? + ??cos3??. (4 Marks) d) Solve the equation ???? ???? = ????2+?? ????2+?? (4 Marks) e) Solve the differential equation ??"(??) - 7??' + 12?? = 0 given that ?? 0 = 2, ??' 0 = 3 (5 Marks) f) Write the series ??3????(?? + 1)??+2 8 ??=-2 as one starting from n=3 instead of n=-2. (4 Marks) g) Verify if the differential equation ???? sin?? + ???? cos?? + ??2 ???? ???? = 0 is exact. (4 Marks)
QUESTION TWO (20 MARKS)
a) Distinguish between a general and an actual solution of a differential equation. (2 Marks) b) Identify the type of each of the following first order linear ordinary differential equations hence solve each of them.
2
i. ?? - ????
?? ?? ???? + ????
?? ?????? = 0 (5 Marks)
ii. ???? ????
= -
2??????+1 ??2????
(4 Marks) iii. ??2???? + 3?????? = 2???? (4 Marks) iv. ?? ???? ???? = ?? + ??3 + 3??2 - 2?? (5 Marks)
QUESTION THREE (20 MARKS) a) i) show that ???? ??2+??2 is a homogeneous function in ?? and ??. (2 Marks) ii) Using the substitution ?? = ????, solve the equation (??2 + ??2)???? = ????????. (5 Marks) b) i) Using an integrating factor to solve the equation 4?? + 3??2 ???? + 2?????? = 0. (5 Marks) ii) Solve the equation ???? ???? + ?? = ????3. (5 Marks) iii) Solve the equation ??2?? ????2 - 4 ???? ???? + 13?? = 0. (3 Marks)
QUESTION FOUR (20 MARKS)
a) The population x of a certain city satisfies the law ???? ????
=
1 100
??, where time t is measured in years. Given that the population of this city is 200,000 in 1980. i. Find the equation connection the population x and time t. (t>1980) ii. Determine in what year the population doubles. (10 Marks) b) i) Obtain the power series solution of ???? ???? = ?? in ascending powers of x. ii) Solve the equation ???? ???? = ?? using the method of separation of variables and verify that the two solutions are equivalent. (10 Marks)
QUESTION FIVE (20 MARKS)
a) Find the general solution of the homogeneous system of equations ???? ???? = 4?? - ?? ???? ???? = ?? + 2?? (10 Marks) b) Consider the linear system. ???? ???? = 5?? + 3?? ???? ???? = 4?? + ?? i. Show that ?? = 3??7??,?? = 2??7?? ?? = ??-??,?? = -2??-?? are solutions of the system.
Prove that the above solutions are linearly independent and write the general solution of the system. (1

Sma2109:Applied Mathematics Question Paper

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