Solve Convolution Method Problems Easily

In summary,The convolution of two functions g andf is the function g "f defined by(g * f)(t) = J>(t - v)f(v)dv .
  • #1
apage
1
0
How do I do this problem?


C. Convolution Method
" - ,c-do..U'."',,..
The convolution of two functions g andfis the function g "f defined by
(g * f)(t) = J>(t - v)f(v)dv .
The aim of this project is to show how convolutions can be used to obtain a particular solution to
a nonhomogeneous equation of the form
(11) ay" + by' + cy = f(t) ,
where a. b. and c are constants, a i=O.


(a) Use Leibniz's rule
d/dt integral from a to t: h(t, v)dv = a to t dh/dt: (t, v)dv + h(t, t) ,
to show that
(Y*f)'(t) = (i *f)(t) -;-y(O)f(t)
and
(y' f)"(t) = (y" *f)(t) -+-y'(O)f(t) + y(O)f'(t) ,
assuming y andf are sufficiently differentiable.
(b) Let y,(t) be the solution to the homogeneous equation ay" + bv' + cv = 0 that satisfies
y,(O) = 0, y~(O) = l/a. Show that Ys* f is the particular solution to equation (II)
satisfyingy(O) = i(O) = o.
(c) Let Yk(t) be the solution to the homogeneous equation av" + by' + C\ = 0 that satisfiesy(
O) = Yo,y'(O) = Y1, and letysbe as defined in part (b). Show that
(Ys*f)(t) +
(12)
is the unique solution to the initial value problem
ay" + by' + cv = f(t); y(O) = Yo ' i(O) = Y1 .
(d) Use the result of part (c) to detennine the solution to each of the following initial value
problems. Carry out all integrations and express your answers in terms of elementary
functions.
(i) + Y = tan t ; y(O) = 0 . )"(0) = -I
(ii) 2y" -+-y' - Y = e -, sin t ; v( 0) = I, y' (0) = I .
(iii) -2y'+y=vte'; v(0)=2, y'(O)=O
i'
 
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  • #2
I couldn't read trough all those symbols, but from what I gather, you're asked to Laplace trasform those equations by definition. Try to look at a Laplace transform book and you're done
 
  • #3
I am doing the project and I am stuck! Did you figure out what to do with part b? or c?
 
  • #4
hmmm...i can't seem to figure out what the heck to do here either, anyone have this solved and could give me some help? i have until midnight tonight...its not looking good!
 

What is the convolution method?

The convolution method is a mathematical technique used to find the output of a system when given its input and a specific mathematical operation to perform on it. It is commonly used in signal processing and image processing applications.

How is the convolution method used to solve problems?

The convolution method involves breaking down a complex problem into smaller, more manageable parts. The input and the operation to be performed on it are represented using mathematical functions, and the output is calculated by integrating the two functions over a specific range.

What types of problems can be solved using the convolution method?

The convolution method can be used to solve a wide range of problems in various fields, including engineering, physics, and mathematics. It is commonly used in solving differential equations, finding the response of linear time-invariant systems, and analyzing signals and images.

What are the benefits of using the convolution method?

The convolution method simplifies complex problems and allows for more efficient and accurate solutions. It also helps to identify the relationship between the input and output of a system, making it a valuable tool in understanding and analyzing various systems.

Are there any limitations to the convolution method?

While the convolution method is a powerful tool, it does have some limitations. It is typically only applicable to linear systems, and it may not be suitable for solving problems with non-linearities. Additionally, it can be computationally intensive for larger systems.

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