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
Use the Laplace transform to solve the following initial value
problem:
x′=12x+3y
y′=−9x+e^(3t)
x(0)=0, y(0)=0
Let X(s)=L{x(t)}, and Y(s)=L{y(t)}.
Find the expressions you obtain by taking the Laplace transform of
both differential equations and solving for Y(s) and X(s):
X(s)=
Y(s)=
Find the partial fraction decomposition of X(s)X(s) and Y(s)Y(s)
and their inverse Laplace transforms to find the solution of the
system of DEs:
x(t)
y(t)
Use the Laplace transform to find the solution of the IVP:
a.) 2y' + y = 1, y(0) = 2 (answer should be y(t) = 1 + e-t
/ 2 )
f.) 4y" + y = 0, y(0) = -1, y'(0) = -1 (answer should be y(t) =
-sin(t) - cos(t))
Please show work!