In your own words - What is the nature of the convolution theorem?

Question
Laplace transform
In your own words - What is the nature of the convolution theorem?

2021-02-04
Step 1
Given
Convolution theorem.
Step 2
The convolution theorem states that,
If $${L}{\left\lbrace f{{\left({t}\right)}}\right\rbrace}={F}{\left({s}\right)}\ \text{ and }\ {L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}={G}{\left({s}\right)}$$ then
$${L}{\left\lbrace f{{\left({t}\right)}}\ast g{{\left({t}\right)}}\right\rbrace}={F}{\left({s}\right)}{G}{\left({s}\right)}\ \text{ where }\ f{{\left({t}\right)}}\ast g{{\left({t}\right)}}={\int_{{0}}^{{t}}} f{{\left({r}\right)}} g{{\left({t}-{r}\right)}}{d}{r}$$
Simply the convolution theorem means if we have two functions then taking their convolution and then Laplace is the same as taking the Laplace first of the two functions separately and then multiplying the two Laplace transforms.
Convolution theorem applied to the solution of differential equation.
Convolution theorem is useful in differentiation and integration of Laplace transforms.

Relevant Questions

Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming.(Write your answer as a function of s.)
$${L}{\left\lbrace{e}^{{-{t}}}\cdot{e}^{t} \cos{{\left({t}\right)}}\right\rbrace}$$
Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.)
$$L\left\{t^2 \cdot te^t\right\}$$
Find the inverse Laplace transform of the given function by using the convolution theorem. $${F}{\left({s}\right)}=\frac{s}{{{\left({s}+{1}\right)}{\left({s}^{2}+{4}\right)}}}$$
Find the inverse Laplace Tranformation by using convolution theorem for the function $$\frac{1}{s^3(s-5)}$$
Find the greatest common divisor (a, b) and integers m and n such that (a, b) = am + bn.
a = 65, b = -91
b.) Explain the meaning of any notation used in the problem and in your solution.
c.) Describe the mathematical concept(s) that appear to be foundational to this problem.
b.)Explain the meaning of any notation used in the problem and in your solution.
c.)Describe the mathematical concept(s) that appear to be foundational to this problem.
d.)Justified solution to or proof of the problem.
Find the greatest common divisior of a,b, and c and write it in the form ax+by+cz for integers x,y, and z.
a=26,b=52,c=60
Use the Laplace transform to solve the following initial value problem:
$$2y"+4y'+17y=3\cos(2t)$$
$$y(0)=y'(0)=0$$
a)take Laplace transform of both sides of the given differntial equation to create corresponding algebraic equation and then solve for $$L\left\{y(t)\right\}$$ b) Express the solution $$y(t)$$ in terms of a convolution integral
In an integro-differential equation, the unknown dependent variable y appears within an integral, and its derivative $$\frac{dy}{dt}$$ also appears. Consider the following initial value problem, defined for t > 0:
$$\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}+{4}{\int_{{0}}^{{t}}}{y}{\left({t}-{w}\right)}{e}^{{-{4}{w}}}{d}{w}={3},{y}{\left({0}\right)}={0}$$
a) Use convolution and Laplace transforms to find the Laplace transform of the solution.
$${Y}{\left({s}\right)}={L}{\left\lbrace{y}{\left({t}\right)}\right)}{\rbrace}-?$$
b) Obtain the solution y(t).
y(t) - ?
$$f*g \text{ of } f(t)=\cos 2t , g(t)=e^t$$