Question

Find the Laplace transform of the given function: f(t)=(t-3)u2(t)-(t-2)u3(t),

where uc(t) denotes the Heaviside function, which is 0 for t<c and 1 for t≥c.

Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).

L{f(t)}= | _________________ , s>0 |

Answer #1

Find the Laplace transform Y(s)=L{y} of the solution of the
given initial value problem.
y′′+9y={t, 0≤t<1 1, 1≤t<∞, y(0)=3, y′(0)=4
Enclose numerators and denominators in parentheses. For example,
(a−b)/(1+n).
Y(s)=

Consider a function (fx) such that L(f)(2) = 1; f(0)=1;
f'(0)=0
Where L(f)(s) denotes the Laplace transform of f(t)
Calculate L(f'')(2)

Find the Laplace transform of the given function. (Express your
answer in terms of s.)
f(t) =
t
3e−(t − τ) sin τ dτ
0

Find the Laplace transform of the function.
(a) f(t) = 2H3 (t) -2H4 (t)
(b) f(t) = t2H3 (t)
(c) Solve x'= -x + H1 (t) - H2 (t), x(0) =
1

Find the Laplace transform of the given function. (Express your
answer in terms of s. Assume that s > 0.)
f(t) =
1,
0 ≤ t < 2
0,
t ≥ 2

Find the inverse Laplace transform L−1{F(s)} of the given
function.
F(s)=(13s2−18s+216)/(s(s2+36))
Your answer should be a function of t.

Write the function in terms of unit step functions. Find the
Laplace transform of the given function.
f(t) =
4,
0 ≤ t < 6
−3,
t ≥ 6

Find the inverse Laplace transform of the given function.
(Express your answer in terms of t.)
F(s) =
8s2 − 10s + 75
s(s2 + 25)

Find the inverse Laplace transform of the given function.
(Express your answer in terms of t.)
F(s) =
8s2 − 8s + 48
s(s2 + 16)

Use Definition 7.1.1,
DEFINITION 7.1.1 Laplace
Transform
Let f be a function defined for
t ≥ 0.
Then the integralℒ{f(t)} =
∞
e−stf(t) dt
0
is said to be the Laplace transform of
f, provided that the integral converges.
to find
ℒ{f(t)}.
(Write your answer as a function of s.)
f(t) = te8t
ℒ{f(t)} =
(s > 8)

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