Higher Engineering Mathematics

Assign-14-H8152.tex 23/6/2006 15: 14 Page 525

I

Differential equations

Assignment 14

This assignment covers the material contained

in Chapters 50 to 53.

The marks for each question are shown in

brackets at the end of each question.

1. Find the particular solution of the following
   differential equations:

(a) 12

d^2 y

dt^2

− 3 y=0 given that whent=0,y= 3

and

dy

dt

=

1

2

(b)

d^2 y

dx^2

+ 2

dy

dx

+ 2 y=10ex given that when

x=0,y=0 and

dy

dx

=1. (20)

2. In a galvanometer the deflectionθsatisfies the
   differential equation:

d^2 θ

dt^2

+ 2

dθ

dt

+θ= 4

Solve the equation forθgiven that whent=0,

θ=0 and

dθ

dt

=0. (12)

3. Determiney(n)wheny= 2 x^3 e^4 x (10)


4. Determine the power series solution of the dif-

ferential equation:

d^2 y

dx^2

+ 2 x

dy

dx

+y=0 using

Leibniz-Maclaurin’s method, given the boundary

conditions that atx=0,y=2 and

dy

dx

= 1. (20)

5. Use the Frobenius method to determine the gen-
   eral power series solution of the differential

equation:

d^2 y

dx^2

+ 4 y= 0 (21)

1

02040 x(cm)

u(x,0)

Figure A14.1

6. Determine the general power series solution of
   Bessel’s equation:

x^2

d^2 y

dx^2

+x

dy

dx

+(x^2 −v^2 )y= 0

and hence state the series up to and including the

term inx^6 whenv=+ 3. (26)

7. Determine the general solution of

∂u

∂x

= 5 xy

(2)

8. Solve the differential equation

∂^2 u

∂x^2

=x^2 (y−3)

given the boundary conditions that atx=0,

∂u

∂x

=sinyandu=cosy. (6)

9. Figure A14.1 shows a stretched string of length
   40 cm which is set oscillating by displacing its
   mid-point a distance of 1 cm from its rest posi-
   tion and releasing it with zero velocity. Solve the

wave equation:

∂^2 u

∂x^2

=

1

c^2

∂^2 u

∂t^2

wherec^2 =1, to

determine the resulting motionu(x,t). (23)