Laplace Transform of Elementary and Special Functions - One Line Questions
1.
Find the Laplace transform of the function f(t) = t^n * sin(bt), where n is a positive integer. —
(-1)^n * d^n/ds^n [b/(s^2 + b^2)]
2.
Calculate the Laplace transform of the function f(t) = t^n * cos(bt), where n is a positive integer. —
(-1)^n * d^n/ds^n [s/(s^2 + b^2)]
3.
What is the Laplace transform of the function f(t) = sin(at+b)? —
(a cos b + s sin b)/(s^2 + a^2)
4.
Determine the Laplace transform of the function f(t) = cos(at+b)? —
(s cos b - a sin b)/(s^2 + a^2)
5.
Find the Laplace transform of the function f(t) = t * cos(at). —
(s^2 - a^2)/(s^2 + a^2)^2
6.
What is the Laplace transform of f(t) = t*cos(bt), for t >= 0? —
(s^2 - b^2)/(s^2 + b^2)^2
7.
Calculate the Laplace transform of the function f(t) = cos(bt) * e^(at), for t >= 0. —
(s-a)/((s-a)^2 + b^2)
8.
What is the Laplace transform of the function f(t) = e^(-at) * cosh(bt), for t >= 0? —
(s+a)/((s+a)^2 - b^2)
9.
What is the Laplace transform of the Dirac delta function, delta(t), assuming delta(t) = 0 for t < 0? —
1
10.
Find the Laplace transform of the unit step function u(t), defined as u(t) = 0 for t < 0 and u(t) = 1 for t >= 0. —
1/s
11.
Find the Laplace transform of the function f(t) = (e^(at) - 1)/(a-1), for t >= 0, assuming a != 1. —
1/((s-1)(s-a))
12.
What is the Laplace transform of the function f(t) = (e^(at) - e^(bt))/ (a-b), for t >= 0? —
1/((s-a)(s-b))
13.
Determine the Laplace transform of the function f(t) = (e^(at) - e^(bt))/(a-b), for t >= 0, assuming a != b. —
1/((s-a)(s-b))
14.
What is the Laplace transform of the function f(t) = (e^(at) - 1)/a, for t >= 0, assuming a is not zero? —
1/(a*s*(s-a))
15.
Determine the Laplace transform of the function f(t) = (t*e^(at))/b, for t >= 0. —
1/(b*(s-a)^2)
16.
Find the Laplace transform of the function f(t) = e^(at) * u(t). —
1/(s-a)
17.
Calculate the Laplace transform of the function f(t) = t*e^(at), for t >= 0. —
1/(s-a)^2
18.
Calculate the Laplace transform of the function f(t) = t * e^(-at). —
1/(s+a)^2
19.
Determine the Laplace transform of the exponential function f(t) = e^(at), for t >= 0, where 'a' is a constant. —
1/(s-a)
20.
Calculate the Laplace transform of the function f(t) = t*u(t). —
1/s^2
21.
Calculate the Laplace transform of the function f(t) = t - sin(bt), for t >= 0. —
1/s^2 - b/(s^2 + b^2)
22.
Determine the Laplace transform of the function f(t) = t^2 * e^(at). —
2/(s-a)^3
23.
What is the Laplace transform of the function f(t) = t^2 * u(t). —
2/s^3
24.
What is the Laplace transform of the function f(t) = t * sin(at)? —
2a*s/(s^2 + a^2)^2
25.
Find the Laplace transform of the function f(t) = t^2 * sin(bt), for t >= 0. —
2b(3s^2 - b^2)/(s^2 + b^2)^3
26.
Find the Laplace transform of the function f(t) = t * sinh(bt). —
2bs/(s^2 - b^2)^2
27.
Determine the Laplace transform of f(t) = t*sin(bt), for t >= 0. —
2bs/(s^2 + b^2)^2
28.
Calculate the Laplace transform of the function f(t) = t^2 * cos(bt), for t >= 0. —
2s(s^2 - 3b^2)/(s^2 + b^2)^3
29.
Determine the Laplace transform of the function f(t) = a*delta(t). —
a
30.
What is the Laplace transform of the function f(t) = t^a, where a is a positive integer? —
a!/s^(a+1)
31.
Find the Laplace transform of the function f(t) = sin(bt) * e^(at), for t >= 0. —
b/((s-a)^2 + b^2)
32.
Determine the Laplace transform of the function f(t) = e^(-at) * sinh(bt), for t >= 0. —
b/((s+a)^2 - b^2)
33.
Find the Laplace transform of the hyperbolic sine function f(t) = sinh(bt), for t >= 0, where 'b' is a constant. —
b/(s^2 - b^2)
34.
Calculate the Laplace transform of the cosine function f(t) = cos(bt), for t >= 0, where 'b' is a constant. —
s/(s^2 + b^2)
35.
Determine the Laplace transform of the hyperbolic cosine function f(t) = cosh(bt), for t >= 0, where 'b' is a constant. —
s/(s^2 - b^2)
36.
Determine the Laplace transform of the function f(t) = sin(bt) * u(t). —
b/(s^2 + b^2)
37.
Find the Laplace transform of the function f(t) = 1 - cos(bt), for t >= 0. —
1/s - s/(s^2 + b^2)
38.
Find the Laplace transform of the sine function f(t) = sin(bt), for t >= 0, where 'b' is a constant. —
b/(s^2 + b^2)
39.
Find the Laplace transform of the function f(t) = delta(t-a), for a > 0. —
e^(-as)
40.
Determine the Laplace transform of the function f(t) = t^a * e^(bt), for t >= 0, where 'a' is a real number greater than -1. —
Gamma(a+1)/(s-b)^(a+1)
41.
Calculate the Laplace transform of the function f(t) = t^a, where 'a' is a real number greater than -1. —
Gamma(a+1)/s^(a+1)
42.
Find the Laplace transform of the Gamma function Gamma(z+1) for z > -1. —
z!/s^(z+1)
43.
What is the Laplace transform of the function f(t) = t^n * e^(at), for t >= 0, where 'n' is a non-negative integer? —
n!/(s-a)^(n+1)
44.
What is the Laplace transform of the function f(t) = t^n, for t >= 0, where 'n' is a non-negative integer? —
n!/s^(n+1)
45.
What is the Laplace transform of the function f(t) = t^n * u(t), where n is a non-negative integer? —
n!/s^(n+1)
46.
What is the Laplace transform of the constant function f(t) = 1, for t >= 0? —
1/s
47.
Determine the Laplace transform of the function f(t) = (a*e^(at) - b*e^(bt)) / (a-b), for t >= 0. —
s/((s-a)(s-b))
48.
Calculate the Laplace transform of the function f(t) = t * cosh(bt). —
s/(s^2 - b^2)^2
49.
Calculate the Laplace transform of the function f(t) = cos(bt) * u(t). —
s/(s^2 + b^2)