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)=
Find the Laplace Transform of the functions
t , 0 ≤ t < 1
(a) f(x) = 2 − t , 1 ≤ t < 2
0 , t ≥ 2
(b) f(t) = 12 + 2 cos(5t) + t cos(5t)
(c) f(t) = t 2 e 2t + t 2 sin(2t)
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 y as a function of t if 9y′′+24y′+32y=0,
y(0)=5,y′(0)=8. y(t)=
2) Find y as a function of x if y′′′+16y′=0,
y(0)=−5, y′(0)=−32, y′′(0)=−32. y(x)=
3) Find y as a function of t if 9y′′−12y′+40y=0,
y(1)=5,y′(1)=9. y=
Find the solution to the heat equation on 0 < x < l,
with u(0, t) = 0, ux(l, t) = 0, and u(x, 0) =
phi(x).
This is sometimes called a "mixed" boundary condition.
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