604 Higher Engineering Mathematics
In Problems 2 to 9, use Laplace transforms to solve
the given differential equations.
- 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
]
- d^2 y
dx^2
- dy
dx
− 2 y=3cos3x−11sin3x,given
y( 0 )=0andy′( 0 )= 6
[y=ex−e−^2 x+sin3x]
- d^2 y
dx^2
− 2
dy
dx
- 2 y=3excos2x,given
y( 0 )=2andy′( 0 )= 5
[
y=3ex(cosx+sinx)−excos2x
]
- Solve, using Laplace transforms, Problems 4
to 9 of Exercise 187, page 480 and Problems
1 to 5 of Exercise 188, page 482. - 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.