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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) i) Distinguish between an ordinary differential equation and a partial differential equation. (2 Marks) ii) Show that ?? = ??2 is a solution of the equation 3??2?? ????2 + 5 ???? ???? = 10?? + 6 . (3 Marks) iii) Form the differential equation from?? = 1 sin3?? + 2 cos3?? by eliminating the constant 1 ?????? 2. (4 Marks) b) i) Show that ???? ??2+??2 is a homogeneous function in x and y. (2 Marks) ii) Using the substitution ?? = ??, transform the equation ???? ???? = ???? ??2+??2 into an equation containing v and x only. (4 Marks) c) consider the differential equation 4?? + 3??2 ???? + 2???????? = 0 i. Show that this equation is not exact. (3 Marks) ii. Find an integrating factor of the form xn where n is a positive integer. (3 Marks) d) Solve the initial value problem ??3?? ????3 - 6 ??2?? ????2 + 11 ???? ???? - 6?? = 0, ?? 0 = 0,?? 0 = 0,??"(0) = 2 . (5 Marks) e) What kind of power series solution exists for the equation ??2 ??2?? ????2 + ?? ???? ???? + ??2 - ??2 ?? = 0 where n is a non-negative constant. (4 Marks)
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QUESTION TWO (20 MARKS)
a) i) Show that the equation 3?? ???? - 2 ???? + ??3 + 2?? ???? = 0 is exact. (3 Marks) ii) Hence solve the equation. (6 Marks) b) i) Define a second order homogeneous linear equation with constant coefficients. (2 Marks) ii) What is a non homogeneous equation of second order? (2 Marks) iii) Solve the following second order homogeneous linear equation with constant coefficients) and show that the solutions are linearly independent ??2?? ????2 - 5 ???? ???? + 6?? = 0. (7 Marks)
QUESTION THREE (20 MARKS)
a) i) Define a second order linear differential equation with variable coefficients. (2 Marks) ii) Consider the equation ??2?? ????2 - 2 ?? ???? ???? + 4 + 2 ??2 ?? = 0. Transform the equation to the form ??2 ????2 + = where = - 1 4 2 - 1 2 ?? ???? and = . Hence solve the equations. (8 Marks)
b) i) Define a linear differential equation of order n. When is this equation said to be homogenous. (2 Marks) ii) Solve the equation ??3?? ????3 - ??2?? ????2 - 4 ???? ???? + 4?? = ??3??. (8 Marks)
QUESTION FOUR (20 MARKS)
Consider the first order systems of equations
???? ????
= 12?? - 15??
???? ????
= 4?? - 4??
i. Write down the equations in matrix form. (2 Marks) ii. Write down the characteristic equation of the matrix. (3 Marks) iii. Calculate the Eigen values of the matrix. (3 Marks) iv. Determine the Eigen vectors corresponding to each of the two Eigen values. (8 Marks) v. Check using the Wronskian whether the two Eigen vectors are independent. (2 Marks) vi. Write the general solution of the system. (2 Marks)
QUESTION FIVE (20 MARKS)
a) i) Obtain the power series solution of the equation ???? ????
= ??. (8 Marks)
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ii) Solve the equation ???? ????
= ?? using the method of separation of variables and verify that the two solutions are equivalent. (4 Marks)
b) 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 800,000 now, i. Find an equation connecting the population x and time t. (5 Marks) ii. After how many years does the population of this city double? (3 Marks)
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