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Math 314: Ordinary Differential Equations I Question Paper
Math 314: Ordinary Differential Equations I
Course:Bachelor Of Education
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION
MATH 314: ORDINARY DIFFERENTIAL EQUATIONS I
STREAMS: BED Y3S1 TIME: 2 HOURS
DAY/DATE: TUESDAY 6/8/2013 2.30 PM – 4.30 PM
INSTRUCTIONS:
Answer Question ONE (Compulsory) and any other TWO Questions
Adhere to the Instructions on the Answer Booklet
Do not Write on the Question Paper.
Question One (30 Marks) Compulsory
(a) Define the following terms [2 marks]
(i) An ordinary differential equation
(ii) A linear ordinary differential equation
(b) State the linearity, order and degree of the following ordinary differential equations giving reasons.
(i) (d^3 y)/?dx?^3 +?4y?^2=0 [2 marks]
(ii) [1+(dy/dx)^2 ]^3=((d^2 y)/?dx?^2 )^2 [2 marks]
(iii) y^''''''+?2e?^x y^''''+?yy?^''=x^4 [2 marks]
(c) Using the method of separation of variables, solve the differential equation.
[3 marks]
dy/dx=?cos?^(2 ) y
(d) Given that dy/dx+2ytanx=sinx evaluate the value of y using an integrating factor.
[4 marks]
(e) Verify that the differential equation
(?5x?^4+?3x?^2 y^2-2xy^3 )dx+(?2x?^3 y-?3x?^2 y^2- 5y^4 )dy=0 is exact hence solve it. [4 marks]
(f) Solve the second order differential equation.
(d^2 y)/?dx?^2 -(6 dy)/dx+9y=0 [9 marks]
(g) Using the method of undetermined coefficients solve for the particular integral of the differential equation. y^''''-3y^''-4y=-?8e?^x cos??2x.? [3 marks]
(h) Using the method of variation of parameters solve the equation.
(d^2 y)/?dx?^2 +y=tan x [5 marks]
Question Two
(a) Given the differential equation
(d^2 y)/?dx?^2 -y=2/(1+e^x ), evaluate
(i) The Auxiliary equation
(ii) Complementary function (cf)
(iii) Using the method of variation of parameters, evaluate the particular integral
(iv) Write the complete solution [10 marks]
(b) Solve the differential equation of y^''''+y=cosecx using the method of variation of parameters. [7 marks]
(c) Solve the differential equation dy/dx=x cosx given that y(0)=0 [3 marks]
Question Three
(a) Given the equation
(d^2 y)/?dx?^2 +?6dy?^2/dx+9y=?5e?^3x
(i) Write the Auxiliary equation and solve it. [1 mark]
(ii) Evaluate the complementary function. [1 mark]
(iii) Evaluate the particular integral, hence, solve the equation. [2 marks]
(b) Solve the differential equation (y^''''-?4y?^''+3y)=x^3 [7 marks]
(c) Determine the particular integral in the equation
(D^2-4D+4) y=x^3 e^2x
Given that D^2=(d^2 y)/(?dx?^x ) ,D=dy/dx [4 marks]
(d) Verify that the differential equation [Cosx ln??(2y-8)+1/x? ]dx+sin?x/(y-4) dy=0
Is exact hence solve it, given that y(1)=9/2 [5 marks]
Question Four
(a) Given the homogeneous linear differential equation
x^2 (d^2 y)/?dx?^2 -2x dy/dx-4y=x^4, reduce it to a differential equation with constant coefficients using appropriate substitutions hence solve it. [8 marks]
(b) Solve the differential equation x^2 (d^2 y)/?dx?^2 +x dy/dx+y=sin?(?logx?^2) using the substitutions x=e^z,x dy/dx=Dy, x^2 (d^2 y)/?dx?^2 =D(D-1)y [8 marks]
(c) Using the method of undetermined coefficients solve y^''''+2y^''+y=x+2
[4 marks]
Question Five
(a) Using the substitution V(x)=y/x, solve the differential equation dy/dx=(x+y)/x [4 marks]
(b) Solve y^''-2xy=x using an appropriate integrating factor. [4 marks]
(c) Solve the equation (d^2 y)/?dx?^2 +µ^2 y=0 [3 marks]
(d) Given the differential equation (d^2 y)/?dx?^2 -2 dy/dx+y=xsinx
Evaluate
(i) Complementary function [3 marks]
(ii) Particular integral [5 marks]
(iii) Complete solution [1 mark]
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