Higher Engineering Mathematics

490 DIFFERENTIAL EQUATIONS

Now try the following exercise.

Exercise 193 Further problems on second

order differential equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy=f(x)wheref(x)is a sum

or product

In Problems 1 to 4, find the general solutions of

the given differential equations.

1. 8

d^2 y

dx^2

− 6

dy

dx

+y= 2 x+40 sinx

⎡

⎣

y=Ae

x

(^4) +Be
x
(^2) + 2 x+ 12

• 8
  17
  (6 cosx−7 sinx)
  ⎤
  ⎦

2. d^2 y
   dθ^2
   − 3
   dy
   dθ

• 2 y=2 sin 2θ−4 cos 2θ
  [
  y=Ae^2 θ+Beθ+^12 ( sin 2θ+cos 2θ)
  ]

3. d^2 y
   dx^2

• dy
  dx
  − 2 y=x^2 +e^2 x
  [
  y=Aex+Be−^2 x−^34
  −^12 x−^12 x^2 +^14 e^2 x
  ]

4. d^2 y
   dt^2
   − 2
   dy
   dt

• 2 y=etsint
  [
  y=et(Acost+Bsint)− 2 tetcost
  ]
  In Problems 5 to 6 find the particular solutions
  of the given differential equations.

5. d^2 y
   dx^2
   − 7
   dy
   dx

• 10 y=e^2 x+20; whenx=0,
  y=0 and
  dy
  dx
  =−
  1
  3
  [
  y=
  4
  3
  e^5 x−
  10
  3
  e^2 x−
  1
  3
  xe^2 x+ 2
  ]

6. 2

d^2 y

dx^2

−

dy

dx

− 6 y=6excosx; when x=0,

y=−

21

29

and

dy

dx

=− 6

20

29

⎡

⎣

y=2e−

3

2 x−2e^2 x

+

3ex

29

(3 sinx−7 cosx)

⎤

⎦