Find the Laplace transform of the following
functions.
(a)
f (t) =
{
6
0 < t ≤ 4
8
t ≥ 4
(b)
f (t) =
{
t2
0 ≤ t < 3
0
t ≥ 3
(c)
f (t) =
{
0
0 ≤ t < π/4
cos[7(t − π/4)]
t ≥ π/4
1. Find the Laplace transform of each of the following
functions: (a). f(t) = t , (b). f(t) = t2 ,
(c) f(t) = tn where n is a positive
integer
Laplace transform of the given function
2. . f(t) = sin bt
3. f(t) = eat sin bt
1) Find the Laplace transform of
f(t)=−(2u(t−3)+4u(t−5)+u(t−8))
F(s)=
2) Find the Laplace transform of f(t)=−3+u(t−2)⋅(t+6)
F(s)=
3) Find the Laplace transform of f(t)=u(t−6)⋅t^2
F(s)=
Solve the following IVP specifically using the Laplace transform
method
(d^3)x/d(t^3)+x=e^(-t)u(t) f(0)=0 f'(0)=0
f''(0)=0
where u(t) is the Heaviside step function
Take the Laplace transform of the following initial value and
solve for X(s)=L{x(t)}X(s)=L{x(t)}:
x′′+16x={sin(πt),0}
0≤t<1
1≤t
x(0)=0
x′(0)=0.
a) X(s)=
Now find the inverse transform to find
b) x(t)=
Use u(t−a) for the Heaviside function shifted a units
horizontaly.