# Convolution Method

1. Jun 29, 2004

### apage

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'

2. Jul 5, 2004

### eJavier

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. Apr 17, 2007

### kls137

I am doing the project and I am stuck! Did you figure out what to do with part b? or c?

4. Oct 30, 2008

### Niner49er52

hmmm...i cant 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!