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Math 403: Ordinary Differential Equations Ii Question Paper
Math 403: Ordinary Differential Equations Ii
Course:Bachelor Of Education Arts
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
FOURTH YEAR EXAMINATIONS FOR THE AWARD OF DEGREE IN BACHELOR OF EDUCATION ARTS
MATH 403: ORDINARY DIFFERENTIAL EQUATIONS II
STREAMS: BED (ARTS) Y4S2 TIME: 2 HOURS
DAY/DATE: WEDNESDAY 17/4/2013 11.30 AM – 1.30PM
INSTRUCTIONS:
Answer questions ONE (compulsory) and any other TWO questions.
QUESTION ONE (COMPULSORY – 30 MARKS)
(a) Given the equation? a?_2 (x) (d^2 y)/(dx^2 )+ a_1 (x) dy/dx+a_0 ?(x)?_y=0. Explain the meaning of the
terms ordinary point and Regular singular point. [2 Marks]
(b) Determine and classify ordinary and singular points of the equation.
x^2 (x-2) (d^2 y)/?dx?^2 +2(x-2) dy/dx+(x+3)_y= 0. [4 Marks]
(c) Show that y_1=sin??3x and y_2=cos?3x ? are a fundamental set of solutions for the
differential equation y^''''=9y=0. [4 Marks]
(d) Use power series method to solve y’’ + y = 0 near x = 0. [5 Marks]
(e) Solve the system of differential equations using the elimination method
x ?=3x-4y
y ?=4x-7y [5 Marks]
(f) Using Bessel’s function relations, show that
(i) (J_0 )^''=?-j?_1 [3 Marks]
(ii) (J_2 )^''=1/2 (J_1-J_3 ) [3 Marks]
(g) Reduce the differential equation into a system of equations and write it in matrix form.
y^''''-3y^''+4y=cos?3t [4 Marks]
QUESTION TWO (20 MARKS)
(a) Using the Taylor’s series expansion, find the power series solution to the initial value
problem.
(d^2 y)/?dx?^2 -xdy/dx-2y=0;y(0)=0,y^'' (0)=1 [5 Marks]
(b) Show that the functions y_1=x^3 and y_2=x^3 In x in x are u independent solution of a2 differential equation. Determine the differential equation. [9 Marks]
(c) (i) Define the gamma function [1 Mark]
(ii) Show that [3 Marks]
(iii) Evaluate [3 Marks]
QUESTION THREE (20 MARKS)
(a) Find the general solution of the system of equations
dx/dt=x-6t
dx/dt=x+2y-z+3
dx/dt=3z-y-10t+4(3z-y-10t+4) [8 Marks]
(b) Classify the analytic points of ?2x?^2 y^''''+6xy^''+(x-1)y=0 [4 Marks]
(c) Show that x = 0 is an ordinary point of the differential equation.
(d^2 y)/?dx?^2 +x^2 y=0 and solve the O.D.E at the point x = 0. [8 Marks]
QUESTION FOUR (20 MARKS)
(a) (i) Distinguish between the Legendre equation and Legendre polynomial.
[3 Marks]
(ii) By expanding(x^2-1)^n and differentiating each term in times; prove that,
p_n (x)=1/2^nn! d^n/?dx?^2 (x^2-1)^n [5 Marks]
(iii) Evaluate p_n (x)=1/(2^n n!) d^n/?dx?^n (x^2-1)^n [7 Marks]
(b) Convert the equation ?2y?^(?^'''' )+3y^''''-4y^''+5y=t^2+16t-20 into an equation of the
form.?(?-y )=A(t) ?(?-y t+?(?-(f(t) y ? ? )=A(t) y ?(t)+f ?(t) )) [5 Marks]
QUESTION FIVE
(a) (i) Define the wronskian of the functions f1, f2, f3 and state its significance. [2 Marks]
(ii) Show that e^(2x ),? e?^(-2x ),e^4x form a basis for the solution of a differential
equation. Write its general solution [4 Marks]
(b) Solve the system of differential equation.
x_1=?2x?_1-?4x?_2+?2t?^2+10t
x^2=x_1-3x_2+t^2+9t+3 [4 Marks]
(c) (i) Define Bessel’s equation and differentiate it form the Bessel’s function.
[2 Marks]
(ii) Express ?4x?^3+?6x?^2+7x+2 in terms of Legendre polynomials. [8 Marks]
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