Higher Engineering Mathematics, Sixth Edition

490 Higher Engineering Mathematics

Now try the following exercise

Exercise 191 Further problems on

differential equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy=f(x)wheref(x)is a sine or

cosine function

In Problems 1 to 3, find the general solutionsof the

given differential equations.

1. 2

d^2 y

dx^2

−

dy

dx

− 3 y=25sin2x

[

y=Ae

3

2 x+Be−x

−^15 (11sin2x−2cos2x)

]

2.

d^2 y

dx^2

− 4

dy

dx

+ 4 y=5cosx

[

y=(Ax+B)e^2 x−^45 sinx+^35 cosx

]

3.

d^2 y

dx^2

+y=4cosx

[y=Acosx+Bsinx+ 2 xsinx]

4. Find the particular solution of the differen-
   tialequation

d^2 y

dx^2

− 3

dy

dx

− 4 y=3sinx;when

x=0,y=0and

dy

dx

=0.

⎡

⎢

⎢

⎣

y=

1

170

(6e^4 x−51e−x)

−

1

34

(15sinx−9cosx)

⎤

⎥

⎥

⎦

5. A differential equation representing the

motion of a body is

d^2 y

dt^2

+n^2 y=ksinpt,

wherek,nandpare constants. Solve the equa-

tion (givenn=0andp^2 =n^2 ) given that when

t=0,y=

dy

dt

=0.

[

y=

k

n^2 −p^2

(

sinpt−

p

n

sinnt

)]

6. The motion of a vibrating mass is given by
   d^2 y
   dt^2

+ 8

dy

dt

+ 20 y=300sin4t. Show that the

general solution of the differential equation is

given by:

y=e−^4 t(Acos 2t+Bsin2t)

+

15

13

(sin 4t−8cos4t)

7. L

d^2 q

dt^2

+R

dq

dt

+

1

C

q=V 0 sinωtrepresents the

variation of capacitor charge in an elec-

tric circuit. Determine an expression for

q at time tseconds given that R= 40 ,

L= 0 .02H, C= 50 × 10 −^6 F, V 0 = 540 .8V

and ω=200rad/s and given the boundary

conditionsthat whent=0,q=0and

dq

dt

= 4. 8

[

q=( 10 t+ 0. 01 )e−^1000 t

+ 0 .024sin200t− 0 .010cos200t

]

51.6 Worked problemson

differential equations of the

forma

d^2 y

dx^2

+b

dy

dx

+cy=f(x)

wheref(x)is a sum or a product

Problem 9. Solve

d^2 y

dx^2

+

dy

dx

− 6 y= 12 x−50sinx.

Using the procedure of Section 51.2:

(i)

d^2 y

dx^2

+

dy

dx

− 6 y= 12 x−50sinx in D-operator

form is

(D^2 +D− 6 )y= 12 x−50sinx

(ii) The auxiliary equation is(m^2 +m− 6 )=0, from

which,

(m− 2 )(m+ 3 )= 0 ,

i.e. m=2orm=− 3

(iii) Since the roots are real and different, the C.F.,

u=Ae^2 x+Be−^3 x.

(iv) Since the right hand side of the given differential

equation is the sum of a polynomial and a sine

function let the P.I.v=ax+b+csinx+dcosx

(see Table 51.1(e)).