Higher Engineering Mathematics, Sixth Edition

604 Higher Engineering Mathematics

In Problems 2 to 9, use Laplace transforms to solve

the given differential equations.

2. 9

d^2 y

dt^2

− 24

dy

dt

+ 16 y=0, giveny( 0 )=3and

y′( 0 )=3.

[

y=( 3 −t)e

4

3 t

]

3.

d^2 x

dt^2

+ 100 x=0, givenx( 0 )=2and

x′( 0 )=0. [x=2cos10t]

4.

d^2 i

dt^2

+ 1000

di

dt

+ 250000 i=0, given

i( 0 )=0andi′( 0 )=100. [i= 100 te−^500 t]

5.

d^2 x

dt^2

+ 6

dx

dt

+ 8 x=0, given x( 0 )=4and

x′( 0 )=8. [x= 4 (3e−^2 t−2e−^4 t)]

6.

d^2 y

dx^2

− 2

dy

dx

+y=3e^4 x,giveny( 0 )=−

2

3

andy′( 0 )= 4

1

(^3) [
y=( 4 x− 1 )ex+
1
3
e^4 x
]
7.
d^2 y
dx^2

• 16 y=10cos4x,giveny( 0 )=3and
  y′( 0 )=4.
  [
  y=3cos4x+sin4x+
  5
  4
  xsin4x
  ]

8. d^2 y
   dx^2

• dy
  dx
  − 2 y=3cos3x−11sin3x,given
  y( 0 )=0andy′( 0 )= 6
  [y=ex−e−^2 x+sin3x]

9. d^2 y
   dx^2
   − 2
   dy
   dx

• 2 y=3excos2x,given
  y( 0 )=2andy′( 0 )= 5
  [
  y=3ex(cosx+sinx)−excos2x
  ]

10. Solve, using Laplace transforms, Problems 4
    to 9 of Exercise 187, page 480 and Problems
    1 to 5 of Exercise 188, page 482.


11. Solve, using Laplace transforms, Problems 3
    to 6 of Exercise 189, page 486, Problems 5
    and 6 of Exercise 190, page 488, Problems 4
    and 7 of Exercise 191, page 490 and Problems
    5 and 6 of Exercise 192, page 492.