Higher Engineering Mathematics, Sixth Edition

Revision Test 15

This Revision Test 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 differ-
   ential 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=10exgiven that whenx=0,

y=0and

dy

dx

=1. (20)

2. In a galvanometer the deflectionθ satisfies the
   differential equation:

d^2 θ

dt^2

+ 2

dθ

dt

+θ= 4

Solve the equation forθ given that whent=0,

θ=0and

dθ

dt

=0. (12)

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


4. Determine the power series solutionof the differen-

tial 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=2and

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)

0 20

1

(^40) x(cm)
u(x,0)
Figure RT15.1

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 RT15.1 shows a stretched string of length
   40cm which is set oscillating by displacing its
   mid-point a distance of 1cm 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

where c^2 =1, to

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