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
+ 16y = 0, given y(0) = 3 and
y (0) = 3.
y = (3 − t ) e
4
3 t
3.
d
2 x
dt 2 + 100x = 0, given x(0) = 2 and
x (0) = 0.
[x = 2 cos10t ]
4.
d
2 i
dt 2 + 1000
di
dt
+ 250000i = 0, given
i(0) = 0 and i (0) = 100. [i = 100t e −500t ]
5.
d 2 x
dt 2 + 6
dx
dt
+ 8x = 0, given x(0) = 4 and
x (0) = 8.
[x = 4(3e −2t − 2e −4t )]
6.
d 2 y
dx 2 − 2
dy
dx
+ y = 3 e
4x , given y(0) = −
2
3
and y (0) = 4
1
3
y = (4x − 1) e x +
1
3
e 4x
7.
d
2 y
dx 2 + 16y = 10 cos4x, given y(0) = 3 and
y (0) = 4.
y = 3 cos4x + sin 4x +
5
4
x sin 4x
8.
d 2 y
dx 2 +
dy
dx
− 2y = 3 cos3x − 11 sin 3x, given
y(0) = 0 and y (0) = 6
[ y = e x − e −2x + sin 3x]
9.
d
2 y
dx 2 − 2
dy
dx
+ 2y = 3 e x cos 2x, given
y(0) = 2 and y (0) = 5
y = 3e x (cos x + sin x) − e x cos 2x
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.